Fractional Mimetic Dark Matter Model

in Fractional Action-like Variational Approach

C. F. L. Godinho and I. V. Vancea

Group of Theoretical Physics and Mathematical Physics, Department of Physics

Federal Rural University of Rio de Janeiro

Cx. Postal 23851, BR 465 Km 7, 23890-000 Seropédica - RJ, Brazil

<EMAIL_ADDRESS><EMAIL_ADDRESS>
27 August 2022

Abstract

In this paper, we propose a new model of fractional mimetic dark matter based
on the fractional action-like variational approach (FALVA) implementation of
fractionality. The model is non-local at classical level and its equations of
motion are highly non-linear. We discuss two particular cases of non-
fractional and fractional mimetic dark fields and solve completely the
equations of motion of these models. In particular, we obtain the metric of
the fractional physical space-time. Also, we obtain the basic geometrical
objects in the two cases.

Keywords: Mimetic Dark Matter Models. Nonlocal Gravity. Fractional Calculus of
Variations. Fractional Gravity. Fractional Field Theory.

##  1 Introduction

The substitution of the standard mathematical formalism used to construct
classical gravitational models, that is differential geometry based on the
standard calculus in flat space-time, by differential objects formulated in
terms of fractional calculus and FALVA formalism to the action functional [1]
\- [6], introduces new phenomena such as nonlocality, memory effects, etc. The
generalization of the Einstein-Hilbert action and the construction of the
fractional differential geometry, including in FALVA formulation was given in
[7] \- [15]. The the fundamental covariant action of fractional gravity was
given for the first time in [14]. These approaches have been used to study
cosmological models in [16]-[33]. Different aspects of the fractional
Newtonian gravity have been discussed in [34]-[42] and the gravitons obtained
from the FALVA model have been examined in [43]. Classical models of gravity
with nonlocal and memory effects have been proposed in [44, 45], too, but
there the nonlocality was the result of the generalization to gravity of the
observation that the measurements in field theory are actually averages over a
time interval (see also the monograph [46]).

In this paper, we propose a new fractional nonlocal gravitational model that
generalizes the ones discussed in the literature and mentioned above by
including a dark matter component. Our proposal is based on the mimetic dark
matter model firstly put forward in [47] in the Lagrangian formulation
developed in [48]-[51] and in the general cosmological background discussed in
[52, 53] and [54]-[57]). What sets apart the mimetic gravity from other
modified theories of gravity is the fact that the dark matter component is a
purely geometrical effect with no matter fields associated to it. In the above
setting, the fields of the model are spatially homogeneous which makes the
model suitable to generalization by the FALVA method. Fractional cosmological
models with this properties in the FALVA approach have been constructed in
[18, 29]. Therefore, the fractioanl mimetic dark matter model (FMDDM) proposed
here is a direct generalization of the models from [18, 29] (see also for
applications of fractional calculus to quantum cosmology [21]-[24] and [25]
for an interesting fractional dark matter model).

Since the nonlocality is introduced by the mathematical formalism, it is
natural to ask what kind of phenomenology FMDDM could be representing. As
discussed above, the nonlocality in the time variable realizes in a new way
the memory effects from [46]. Also, since the FALVA formalism contains terms
that can be interpreted, at least in some fractional mechanical systems, as
nonconservative forces, the FMDDM realizes a new type of nonconservative
gravity [58, 59] In order to implement the fractionality, we use the
formulation of the FMDDM in terms of the degrees of freedom of the auxiliary
metric. From that, it follows that the physical space-time geometry inherits
fractionality properties which are reflected in phenomenological parameters
that can be used to test fractional structure.

The paper is organized as follows. In Section 2, we briefly review the mimetic
dark matter model. In Section 3, we present the FMDMM in the FALVA approach.
Here, we give the action and derive the Euler-Lagrange equations. In this
section, we also solve the Euler-Equations for a background characterized by a
constant auxiliary Hubble parameter. We obtain the general solution
parametrized by the fractionality parameter. In Section 4, we calculate the
physical Ricci tensor corresponding to the general solution and discuss the
cosmological redshift and the cosmological distances in two limits for
different values of the fractionality parameter. We drive our conclusions in
the last section. Throughout this work, we use the space-time metric with
signature $(-,+,+,+)$ and the natural units with $c=1$.

##  2 Mimetic Dark Matter

In this section, we briefly review the formulation of the mimetic dark matter
model. The results presented here can be found in the original works [48]-[51]
which we refer to for further details.

The mimetic gravity is a generalization of the General Relativity in which the
physical space-time metric $g_{\mu\nu}$ is parametrized in terms of the
conformal scalar degree of freedom $\phi$, also called mimetic field. This is
realized by introducing an auxiliary metric $l_{\mu\nu}$ which is related to
$g_{\mu\nu}$ and $\phi$ by the following equation

$g_{\mu\nu}:=\left(l^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\right)l_{\mu\nu}=\Phi^{2}l_{\mu\nu}\,.$
(1)

In this setting, $g_{\mu\nu}$ is manifestly invariant under the Weyl
transformations of the auxiliary metric

$l_{\mu\nu}\to{l^{\prime}}_{\mu\nu}=\omega^{2}l_{\mu\nu}\,,$ (2)

where $\omega=\omega(x)$ is an arbitrary function on the coordinates. A
straightforward consistency condition follows from the above equation

$\Phi^{2}=l^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,.$ (3)

The dynamics of the mimetic gravity is governed by the Einstein-Hilbert action
expressed in terms of the field variables $l_{\mu\nu}$ and $\phi$

$S_{0}[l_{\mu\nu},\phi]=-\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g(l_{\mu\nu},\phi)}\,R\left[g_{\mu\nu}(l_{\mu\nu},\phi)\right)]\,,$
(4)

where $\kappa=8\pi G$. One can show that the action (4) takes the following
form [47]

$S_{1}[l_{\mu\nu},\phi,\Phi,\lambda]=-\frac{1}{2\kappa}\int
d^{4}x\sqrt{-l}\left[\Phi^{2}R\left(l_{\mu\nu}\right)+6l^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\lambda\left(\Phi^{2}-l^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right)\right]\,,$
(5)

where $l^{\mu\nu}$ denotes the inverse of the auxiliary metric

$l^{\mu\nu}l_{\nu\rho}={\delta^{\mu}}_{\rho}\,.$ (6)

The last term from the action (5) implements the constraint that represents
the consistency condition (3) and $\lambda$ is the corresponding Lagrange
multiplier [48, 49]. Although it is possible to formulate the Einstein-Hilbert
action without the constraint, the equation (5) is convenient for
generalization to a FALVA model since it provides an unique procedure to
extend both the action and the metric consistency constraint to the fractional
fields. The mimetic gravity model has three degrees of freedom: two transverse
polarizations of the gravitational field and one conformal degree of freedom
of the scalar field.

In what follows, we are going to consider the general homogeneous FLRW
cosmological background characterized by the following line element

$d^{2}s_{g}=\mathcal{N}^{2}(t)dt^{2}-\mathcal{A}^{2}(t)\delta_{ij}dx^{i}dx^{j}\,.$
(7)

The physical metric tensor corresponding to $d^{2}s_{g}$ can be obtained from
non-vanishing spatially homogeneous fields $\phi(t)$ and $l_{\mu\nu}(t)$
which, at their turn, define the following auxiliary line element

$d^{2}s_{l}=N^{2}(t)dt^{2}-a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,.$ (8)

The relation between the physical and auxiliary metrics follows from the
definition (1) which is equivalent to the following equations

$\mathcal{N}^{2}(t)=\Phi^{2}(t)N^{2}(t)\,,\qquad\mathcal{A}^{2}(t)=\Phi^{2}(t)a^{2}(t)\,,\qquad\Phi^{2}(t)=N^{-2}(t)\dot{\phi}^{2}(t)\,.$
(9)

By substituting the equations $d^{2}s_{g}$ and $d^{2}s_{l}$ into the action
(5) the action $S_{1}[l_{\mu\nu},\phi,\Phi,\lambda]$ can be put into the
following form

$S_{1}[N,a,\phi,\Phi,\lambda]=-\frac{1}{2\kappa}\int
dtNa^{3}\left[6\Phi^{2}\left(\frac{\ddot{a}}{N^{2}a}+\frac{\dot{a}^{2}}{N^{2}a^{2}}-\frac{\dot{N}\dot{a}}{N^{3}a}\right)+\frac{6\dot{\Phi}^{2}}{N^{2}}-\lambda\left(\Phi^{2}-\frac{\dot{\phi}^{2}}{N^{2}}\right)\right]\,.$
(10)

Note that the last equation from (9) is just the consistency constraint (3)
from above. As before, the constraint is specified by the equation of motion
of the Lagrange multiplier $\lambda$. The relevant variable here is time $t$
since the spatial integration generates an overall constant volume factor that
plays no role in the dynamics of fields.

It is worth mentioning that the density energy of the scalar field in the FLRW
background goes with $\mathcal{A}^{-3}$. The dependence of the density energy
on $\mathcal{A}$ is not affected by the presence of other matter fields. Thus,
the mimetic gravity model has properties similar to the cold dark matter. For
more details on the application of this model to cosmology, see e. g. [52,
53].

##  3 Fractional Mimetic Dark Matter Models in FALVA Formulation

In this section, we generalize the mimetic dark matter model discussed in the
previous section to the fractional gravity. We are going to use the FALVA
formulation without fractional derivatives for which we follow [1]-[5] and we
abbreviate this model by FALVA-FMDDM. Firstly, we are going to give the most
general FALVA-FMDDM. Then we are going to discuss two particular models
constructed in backgrounds of higher symmetry determined by particular forms
of the auxiliary metric. The first particular FALVA-FMDDM has a standard non
fractional mimetic field. For the second model, we consider a simple ansatz
for the fractional mimetic field.

###  3.1 General FALVA-FMDMM

The basic object in the FALVA formulation of the fractional field theory is
the fractional action integral. For a set of fields $\psi$ in one-dimension,
the fractional action takes the following form

$S^{\alpha}[\psi](t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{a}d\tau
L(\psi(\tau),\dot{\psi}(\tau),\ddot{\psi}(\tau),\tau)(t-\tau)^{\alpha-1}\,,$
(11)

where $\psi:[a,b]\in\mathbb{R}\rightarrow M$ satisfies the boundary conditions
$\psi(a)=\psi_{1}$ and $\psi(b)=\psi_{2}$, $\alpha$ is the fractionality
parameter $\alpha\in(0,1]$ and the dot denotes the standard derivative with
respect to $\tau$ variable. Here,
$\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}\exp(-t)\,dt$ is the Euler gamma
function. Alternatively, the action $S^{\alpha}(t)$ can be written as follows

$S^{\alpha}[\psi](t)=\int_{a}^{t}L(\psi(\tau),\dot{\psi}(\tau),\ddot{\psi}(\tau),\tau)d\mu_{t}\,,$
(12)

where
$\mu_{t}(\tau)=\frac{{1}}{{\Gamma(\alpha+1)}}\\{t^{\alpha}-(t-\tau)^{\alpha}\\}$
has a scale property that generates a memory effect. Typically, the complexity
of physical systems is connected to the memory of long range interactions by
non-local (spatial) effects which are incorporated in the FALVA formalism
through the integration measure $d\mu_{t}$ [61].

We define the FALVA-FMDMM by generalizing $S_{1}[N,a,\phi,\Phi,\lambda]$ from
the equation (10) to the following fractional action integral

$\displaystyle S^{\alpha}[N,a,\phi,\Phi,\lambda](t)$
$\displaystyle=-\frac{1}{2\kappa\Gamma(\alpha)}\int^{t}_{a}d\tau
Na^{3}\left[6\Phi^{2}\left(\frac{\ddot{a}}{N^{2}a}+\frac{\dot{a}^{2}}{N^{2}a^{2}}-\frac{\dot{N}\dot{a}}{N^{3}a}\right)+\frac{6\dot{\Phi}^{2}}{N^{2}}\right.$
$\displaystyle-\left.\lambda\left(\Phi^{2}-\frac{\dot{\phi}^{2}}{N^{2}}\right)\right](t-\tau)^{\alpha-1}\,.$
(13)

The last term from $S^{\alpha}[N,a,\phi,\Phi,\lambda]$ implements the metric
constraint in the fractional model. The derivatives of the fields
$\psi=\left\\{a,N,\Phi,\lambda,\phi\right\\}$ are defined by the standard
calculus as required by the FALVA construction. One can also interpret the
equation (13) as describing a set of models parametrized by the fractionality
parameter $\alpha$.

The dynamics of the FALVA-FMDMM is described by the fractional variational
principle applied to $S^{\alpha}[N,a,\phi,\Phi,\lambda]$ with the variation
taken over the space of the functions $\psi$ with fixed end points. Then the
variational principle is equivalent to the fractional Euler-Lagrange equations
that take the following form [5]

$\displaystyle\mathcal{D}\psi$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\psi}-\frac{d}{d\tau}\left(\frac{\partial\mathcal{L}}{\partial\dot{\psi}}\right)+\frac{d^{2}}{d\tau^{2}}\left(\frac{\partial\mathcal{L}}{\partial\ddot{\psi}}\right)$
$\displaystyle+\left(\frac{1-\alpha}{t-\tau}\right)\frac{\partial\mathcal{L}}{\partial\dot{\psi}}-2\left(\frac{1-\alpha}{t-\tau}\right)\frac{d}{d\tau}\left(\frac{\partial\mathcal{L}}{\partial\ddot{\psi}}\right)+\frac{(1-\alpha)(2-\alpha)}{(t-\tau)^{2}}\frac{\partial\mathcal{L}}{\partial\ddot{\psi}}=0\,,$
(14)

By applying the equation (14) to the fields of the model, we obtain the
following equations

$\displaystyle\mathfrak{D}a$
$\displaystyle=3\Phi^{2}N^{2}[2aN(\dot{a}\dot{N}-\ddot{a})-N\dot{a}^{2}]+\dot{N}-\dot{\Phi}N-\Phi\dot{N}-6aN^{3}\Phi(\ddot{a}\Phi+a\ddot{\Phi})$
$\displaystyle+6a^{2}\Phi^{2}\dot{N}N+6N^{3}\Phi^{2}(\dot{a}^{2}+\ddot{a}a-2\dot{a})+3\left(\dfrac{1-\alpha}{t-\tau}\right)N^{3}\Phi\left[4a(\dot{a}\Phi+a\dot{\Phi})-2\Phi\dot{a}\right]$
$\displaystyle-\dfrac{3(1-\alpha)(2-\alpha)}{(t-\tau)^{2}}a^{2}\Phi^{2}N^{3}=0\,,$
(15) $\displaystyle\mathfrak{D}N$
$\displaystyle=\Phi(\ddot{a}a+\dot{a}^{2}a)-2\dot{\Phi}N\dot{a}a^{2}-aN\Phi(\ddot{a}a+2\dot{a}^{2})-12\Phi^{3}-6\lambda\Phi\phi^{2}$
$\displaystyle+\left(\dfrac{1-\alpha}{t-\tau}\right)\dot{a}a^{2}\Phi N=0\,,$
(16) $\displaystyle\mathfrak{D}\Phi$
$\displaystyle=\Phi[a(\dot{N}\dot{a}+N\ddot{a})-N\dot{a}^{2}]+aN[3\dot{a}\dot{\Phi}+a\ddot{\Phi}-\dot{N}a\Phi]-\left(\dfrac{1-\alpha}{t-\tau}\right)a^{2}\Phi
N^{3}=0\,,$ (17) $\displaystyle\mathfrak{D}\lambda$
$\displaystyle=\Phi^{2}-\frac{\dot{\phi}^{2}}{N^{2}}=0\,,$ (18)
$\displaystyle\mathfrak{D}\phi$ $\displaystyle=6\lambda\dot{a}\dot{\phi}\phi
N+\lambda a\ddot{\phi}\phi N+\lambda
a\dot{\phi}^{2}N+\dot{\lambda}a\dot{\phi}\phi N-\lambda
a\dot{\phi}\phi\dot{N}+\left(\dfrac{1-\alpha}{t-\tau}\right)\lambda
a\dot{\phi}\phi N=0\,.$ (19)

Note that the non-linear differential equations (15) - (19) are defined in the
standard calculus. These equations are non-local and they contain damping
coefficients that determine the field dynamics. The equation (18) is the
constraint on $\Phi$ which is a consequence of the metric structure of the
original mimetic model that is assumed to hold in the fractional case.

###  3.2 FALVA-FMDMM with non-fractional mimetic field

In general, the set of non-linear equations (15) - (19) is difficult to solve.
However, some simplifications can be made for models with specific properties,
e. g. for higher symmetric backgrounds. As an example, consider the
backgrounds with constant auxiliary Hubble parameter $\dot{a}/a=H_{0}$ and
homogeneous field $\phi(x)=\phi(t)$. By implementing these properties at the
Lagrangian level and after solving the consistency condition which fixes
$\phi(t)=t+C$, where $C$ is a constant which we set to zero, we obtain the
following Lagrangian

$\mathcal{L}=-\frac{3\,a^{3}}{\kappa\Gamma(\alpha)N^{3}}\left[H_{0}\left(2H_{0}-\frac{\dot{N}}{N}\right)+\frac{\dot{N}^{2}}{N^{2}}\right]\left(t-\tau\right)^{\alpha-1}\,.$
(20)

The degrees of freedom of this model are the fields $a(\tau)$ and $N(\tau)$
with $\phi=\tau$. The field $a(\tau)$ does not enter the Lagrangian
dynamically. However, its form is fixed by the auxiliary Hubble parameter
condition to $a(\tau)=\exp(H_{0}\tau)$. Since $a(\tau)$ is determined by a
condition which is not an equation of motion, we can use the exponential
function to fix the form of the Lagrangian further to

$\mathcal{L}=-\frac{3\,e^{3H_{0}\tau}\left(t-\tau\right)^{\alpha-1}}{\kappa\Gamma(\alpha)N^{3}}\left[H_{0}\left(2H_{0}-\frac{\dot{N}}{N}\right)+\frac{\dot{N}^{2}}{N^{2}}\right]\,.$
(21)

The equation of motion $\mathcal{D}N=0$ can be obtained by using the formula
(14). After some calculations, we are led to the following non-linear first
order differential equation

$\mathcal{D}N=2\frac{\ddot{N}}{N}-3\frac{\dot{N}^{2}}{N^{2}}+\left(\frac{1-\alpha}{t-\tau}+8H_{0}-6\right)\frac{\dot{N}}{N}-\left(\frac{1-\alpha}{t-\tau}+3H^{2}_{0}-6H_{0}\right)=0\,.$
(22)

By making the substitution $F(\tau;\alpha)=\dot{N}/N$, the equation (22) is
transformed into the following Riccati equation

$\dot{F}(\tau;\alpha)-\frac{1}{2}F^{2}(\tau;\alpha)+\left(\frac{1-\alpha}{t-\tau}+8H_{0}-6\right)F(\tau;\alpha)-\frac{1}{2}\left(\frac{1-\alpha}{t-\tau}+3H^{2}_{0}-6H_{0}\right)=0\,.$
(23)

The equation (23) can be converted into a second-order linear ordinary
differential equation by introducing the following function

$u(\tau;\alpha)=\exp\left(-\frac{1}{2}\int d\tau F(\tau;\alpha)\right)\,.$
(24)

After substituting the equation (24) in to the equation (23) and after some
algebraic manipulations we obtaine the following final equation

$\displaystyle\ddot{u}-\left(\frac{1-\alpha}{\tau-t}-8H_{0}+6\right)\dot{u}+\left(\frac{\alpha-1}{\tau-t}+3H^{2}_{0}-6H_{0}\right)=0\,.$
(25)

The general solution of the equation (25) can be found by standard methods and
it is has the form

$\displaystyle u(\tau;\alpha)$
$\displaystyle=\int^{\tau}_{1}d\xi\,\left\\{\frac{e^{2(4H_{0}-3)(t-\xi)}}{2^{\alpha+2}}\left(t-\xi\right)\left[2\left(\alpha-1\right)\left(4H_{0}-3\right)\Gamma\left(\alpha-1,2(4H_{0}-3)(t-\xi)\right)\right.\right.$
$\displaystyle-\left.\left.3\left(H_{0}-2\right)H_{0}\Gamma\left(\alpha,2(4H_{0}-3)(t-\xi)\right)\right]\left[\left(4H_{0}-3\right)\left(t-\xi\right)\right]^{-\alpha}\right.$
$\displaystyle+\left.e^{2(4H_{0}-3)(t-\xi)-(\alpha-1)\ln(t-\xi)}C_{1}\right\\}+C_{2}\,.$
(26)

Here, $\Gamma(a,x)$ is the incomplete gamma function

$\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}\,e^{-t}\,dt,\,,$ (27)

and $C_{1}$ and $C_{2}$ are integration constants. The solution of the
equation $\mathcal{D}N=0$ is written as

$N(\tau;\alpha)=u^{-2}(\tau;\alpha)\,.$ (28)

The above equation shows that the function $u(\tau;\alpha)$ determines a
family of fractional auxiliary backgrounds parametrized by the parameter
$\alpha$.

Let us discuss the function $u(\tau;\alpha)$. To simplify the formulas, we
introduce the variable $z=2(4H_{0}-3)(t-\xi)$ which is positive for $t>\xi$
and $H_{0}>3/4$ or $t<\xi$ and $H_{0}<3/4$. At $H_{0}=3/4$, the change of
variable $\xi\to z$ is not well defined. The equation (26) takes the following
form in terms of $z$ variable

$\displaystyle u(\tau;\alpha)$
$\displaystyle=-\frac{1}{2(4H_{0}-3)}\int^{b}_{a}dz\,\left\\{\frac{e^{z}z^{1-\alpha}}{8(4H_{0}-3)}\left[2\left(\alpha-1\right)\left(4H_{0}-3\right)\Gamma\left(\alpha-1,z\right)\right.\right.$
$\displaystyle-\left.\left.3\left(H_{0}-2\right)H_{0}\Gamma\left(\alpha,z\right)\right]+C_{1}e^{z}\left[\frac{z}{2(H_{0}-3)}\right]^{1-\alpha}\right\\}+C_{2}\,,$
(29)

where the integration limits are $a=2(4H_{0}-3)(t-1)$ and
$b=2(4H_{0}-3)(t-\tau)$ and $C_{1}$ and $C_{2}$ are real constants.

At small values of $z$, one can approximate
$\Gamma(\alpha,z)/z^{\alpha}\sim-\frac{1}{\alpha}$ since $\alpha<1$ [60]. Then
$u(\tau;\alpha)$ is given by the following formula at lowest order in $z$

$\displaystyle u_{0}(\tau,\alpha)$
$\displaystyle\simeq\frac{e^{2(4H_{0}-3)(t-1)}}{8(4H_{0}-3)}\left\\{\exp\left(\frac{t-\tau}{t-1}\right)-1-\frac{3(H_{0}-2)H_{0}}{2(4H_{0}-3)\alpha}\left[\exp\left(\frac{t-\tau}{t-1}\right)2(4H_{0}-3)(t-\tau)\right.\right.$
$\displaystyle-\left.\left.\exp\left(\frac{t-\tau}{t-1}\right)-2(4H_{0}-3)(t-1)+1\right]\right\\}+C\,.$
(30)

where $C$ is a constant.

Another approximation can be made at large values of $z$ where the equation
(26) is approximated by the following one

$\displaystyle u_{1}(\tau;\alpha)$
$\displaystyle\simeq-\frac{1}{8(4H_{0}-3)}\ln\left(\frac{t-\tau}{t-1}\right)-\frac{3(H_{0}-2)H_{0}}{8(4H_{0}-3)}\left(1-\tau\right)$
$\displaystyle-
C_{1}2^{\alpha-6}(4H_{0}-3)^{2\alpha-4}\left[e^{2(4H_{0}-3)(t-\tau)}(t-\tau)^{\alpha-1}-e^{2(4H_{0}-3)(t-1)}(t-1)^{\alpha-1}\right]+C_{2}\,,$
(31)

with $C_{1}$ and $C_{2}$ are constants. However, one should note that large
$z$ is equivalent with large $t-\xi$ and $\xi\in[1,\tau]$ with $\tau\in[1,t]$.
Recall that the initial action is defined at a fixed value of $t$, therefore
the formula (31) is interesting in the limit
$\lim_{t\to\infty}S^{\alpha}[\psi](t)$.

###  3.3 FALVA-FMDMM with a fractional mimetic field

The FALVA-FMDMM from the previous section has a non-fractional field $\phi$
which is the same as in the standard mimetic dark model. The simplest model
with fractional mimetic field can be obtained by taking the following ansatz
$\phi(t)=t^{\alpha+1}$. Since the constraint (3) is the same, it follows that

$\mathcal{N}(t)=\pm(\alpha+1)t^{\alpha}\,.$ (32)

Thus, we can take $\Phi(t)=\pm(\alpha+1)t^{\alpha}N^{-1}(t)$ which leaves the
field $N$ the only degree of freedom of the model. Its dynamics follows from
the Lagrangian

$\mathcal{L}=-\frac{3\left(\alpha+1\right)^{2}}{\kappa\Gamma(\alpha)}\left(\frac{e^{H_{0}\tau}}{N}\right)^{3}\left[-H_{0}\frac{\dot{N}}{N}+\frac{\alpha^{2}}{\tau^{2}}+2H^{2}_{0}\right]\tau^{2\alpha}(t-\tau)^{\alpha-1}\,.$
(33)

By applying the fractional variational principle, we obtain the following
equation of motion

$\frac{\dot{N}}{N}-\frac{3\alpha^{2}}{H_{0}\tau^{2}}-\frac{2\alpha}{\tau}+6H_{0}-3=0\,.$
(34)

This is an ordinary differential equation that has the following solution

$N(\tau;\alpha)=C_{3}\,\tau^{-2\alpha}\exp\left[-\frac{3\alpha}{H_{0}\tau}+3(2H_{0}-1)\tau\right]\,,$
(35)

where $C_{3}$ is a real integration constant. Then the conformal factor is

$\Phi(\tau;\alpha)=\pm
C_{3}\left(\alpha+1\right)\tau^{-\alpha}\exp\left[-\frac{3\alpha}{H_{0}\tau}+3(2H_{0}-1)\tau\right]\,.$
(36)

The solutions of the equations of motion obtained above allow one to explore
the physical properties of these models as well as the consequences of the
fractionality on the properties of space-time, which we are going to do in the
next section.

##  4 Geometrical Properties of FALVA-FMDMMs

As we have seen in the previous section, the fractional character of the
FMDMMs is defined primarily by the fractional gravity in the FALVA
construction. As such, we expect that the geometry of the space-time be
fractional in the sense that it should depend on the fractionality parameter.
Another source of fractionality of geometry is the mimetic field which
determines the physical metric. Above, we have seen that at least for two
simplified models with non-fractional and fractional mimetic fields,
respectively, the dynamics of the degrees of freedom can be completely
determined. This is possible since the equations of motion become tractable at
the cost of additional assumptions on the fields.

In this section, we are going to investigate some geometrical and physical
properties of these two models. We focus mainly on the fractional aspect of
the geometrical tensors which are defined by the standard differential
calculus. These definitions need not be modified since the FALVA construction
is based on the standard calculus. Therefore, we can say that, in a general
sense, the geometries of each model belong to new classes of geometries, the
novelty being the parametrization by the fractionality parameter.

###  4.1 Geometry of FALVA-FMDMM with non-fractional mimetic field

The mimetic field of the model given in subsection 3.2 is the same as in the
mimetic dark matter extension of the General Relativity. By substituting the
equation (28) into the metric, we conclude that the mimetic field $\phi$
induces the following space-time metric

$ds^{2}_{g}=-dt^{2}+e^{2H_{0}t}u^{4}(\tau;\alpha)\delta_{ij}dx^{i}dx^{j}\,,$
(37)

where the general form of $u(\tau;\alpha)$ is given by the equation (29). From
the equation (37), we can derive the main geometrical objects of the family of
fractional space-time geometries parametrized by $\alpha$. In order to
simplify the notation, we drop the time and fractionality parameter from
formulas.

The non-vanishing Christoffel symbols have the following form

$\displaystyle{\Gamma^{0}}_{11}$
$\displaystyle=e^{2H_{0}t}u^{3}\left(H_{0}u+2\dot{u}\right)\,,$ (38)
$\displaystyle{\Gamma^{0}}_{22}$
$\displaystyle=e^{2H_{0}t}u^{3}\left(H_{0}u+2\dot{u}\right)\,,$ (39)
$\displaystyle{\Gamma^{0}}_{33}$
$\displaystyle=e^{2H_{0}t}u^{3}\left(H_{0}u+2\dot{u}\right)\,,$ (40)
$\displaystyle{\Gamma^{1}}_{10}$ $\displaystyle=H_{0}+\frac{2\dot{u}}{u}\,,$
(41) $\displaystyle{\Gamma^{2}}_{20}$
$\displaystyle=H_{0}+\frac{2\dot{u}}{u}\,,$ (42)
$\displaystyle{\Gamma^{3}}_{30}$ $\displaystyle=H_{0}+\frac{2\dot{u}}{u}\,.$
(43)

By using the Christoffel symbols, we can calculated the components of the
Riemann tensor and we obtain the following results

$\displaystyle{R^{0}}_{110}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H^{2}_{0}u^{2}+2\dot{u}^{2}+2u\left(2H_{0}\dot{u}+\ddot{u}\right)\right)\,,$
(44) $\displaystyle{R^{0}}_{220}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H^{2}_{0}u^{2}+2\dot{u}^{2}+2u\left(2H_{0}\dot{u}+\ddot{u}\right)\right)\,,$
(45) $\displaystyle{R^{0}}_{330}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H^{2}_{0}u^{2}+2\dot{u}^{2}+2u\left(2H_{0}\dot{u}+\ddot{u}\right)\right)\,,$
(46) $\displaystyle{R^{1}}_{010}$
$\displaystyle=-H^{2}_{0}-\frac{2\dot{u}^{2}}{u^{2}}-\frac{2\left(2H_{0}\dot{u}+\ddot{u}\right)}{u}\,,$
(47) $\displaystyle{R^{1}}_{221}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,,$ (48)
$\displaystyle{R^{1}}_{331}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,,$ (49)
$\displaystyle{R^{2}}_{020}$
$\displaystyle=-H^{2}_{0}-\frac{2\dot{u}^{2}}{u^{2}}-\frac{2\left(2H_{0}\dot{u}+\ddot{u}\right)}{u}\,,$
(50) $\displaystyle{R^{2}}_{121}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,,$ (51)
$\displaystyle{R^{2}}_{332}$
$\displaystyle=-e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,,$ (52)
$\displaystyle{R^{3}}_{030}$
$\displaystyle=-H^{2}_{0}-\frac{2\dot{u}^{2}}{u^{2}}-\frac{2\left(2H_{0}\dot{u}+\ddot{u}\right)}{u}\,,$
(53) $\displaystyle{R^{3}}_{131}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,,$ (54)
$\displaystyle{R^{3}}_{232}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(H_{0}u+2\dot{u}\right)^{2}\,.$ (55)

The next tensor of interest is the Ricci tensor. Its non-vanishing components
are

$\displaystyle R_{00}$
$\displaystyle=-3H^{2}_{0}-\frac{6\dot{u}^{2}}{u^{2}}-\frac{6\left(2H_{0}\dot{u}+\ddot{u}\right)}{u}\,,$
(56) $\displaystyle R_{11}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(3H^{2}_{0}u^{2}+10\dot{u}^{2}+2u\left(6H_{0}\dot{u}+\ddot{u}\right)\right)\,,$
(57) $\displaystyle R_{22}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(3H^{2}_{0}u^{2}+10\dot{u}^{2}+2u\left(6H_{0}\dot{u}+\ddot{u}\right)\right)\,,$
(58) $\displaystyle R_{00}$
$\displaystyle=e^{2H_{0}t}u^{2}\left(3H^{2}_{0}u^{2}+10\dot{u}^{2}+2u\left(6H_{0}\dot{u}+\ddot{u}\right)\right)\,.$
(59)

It is easy to see that the Ricci scalar has the following form

$R=\frac{12\left[H^{2}_{0}u^{2}+3\dot{u}^{2}+u\left(4H_{0}\dot{u}+\ddot{u}\right)\right]}{u^{2}}\,.$
(60)

The geodesic equations can be calculated from the Christoffels symbols. To
this end, we introduce the components of the velocity
$v^{\mu}(s)=dx^{\mu}(s)/ds$ along the geodesics as functions on the proper
time $s$. Then we get the following set of equations

$\displaystyle\frac{dv^{0}}{ds}$
$\displaystyle=-e^{2H_{0}t}\left[(v^{1})^{2}+(v^{2})^{2}+(v^{3})^{2}\right]u^{4}\left(H_{0}+\frac{2\dot{u}}{u}\right)\,,$
(61) $\displaystyle\frac{dv^{1}}{ds}$
$\displaystyle=-2v^{0}v^{1}\left(H_{0}+\frac{2\dot{u}}{u}\right)\,,$ (62)
$\displaystyle\frac{dv^{2}}{ds}$
$\displaystyle=-2v^{0}v^{2}\left(H_{0}+\frac{2\dot{u}}{u}\right)\,,$ (63)
$\displaystyle\frac{dv^{3}}{ds}$
$\displaystyle=-2v^{0}v^{3}\left(H_{0}+\frac{2\dot{u}}{u}\right)\,.$ (64)

The above relations have been obtained for the general $u(\tau;\alpha)$. When
the explicit time-dependence of $u$ on $\tau$ is plugged-in, concrete, but not
very illuminating formulas, are obtained.

###  4.2 Geometry of FALVA-FMDMM with fractional mimetic field

The FALVA-FMDMM with fractional mimetic field given by the Lagrangian (33)
induces the following fractional metric in the physical space-time

$ds^{2}_{g}=(1+\alpha)^{2}t^{2\alpha}\left[-dt^{2}+C_{3}^{-2}t^{4\alpha}e^{(6-10H_{0})t+\frac{6\alpha}{H_{0}t}}\,\delta_{ij}dx^{i}dx^{j}\right]\,.$
(65)

Here, we have express the results in terms of the physical time. For
simplicity of notation, we omit the dependence on time and fractionality
parameter in the components of the computed quantities and we use the
Cartesian coordinates.

As in the previous case, the first geometrical object to be computed are the
Christoffel symbols ${\Gamma^{\mu}}_{\nu\rho}$ corresponding to the metric
(65). The non-vanishing components of ${\Gamma^{\mu}}_{\nu\rho}$ are

$\displaystyle{\Gamma^{0}}_{00}$ $\displaystyle=\frac{\alpha}{t}\,,$ (66)
$\displaystyle{\Gamma^{0}}_{11}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-1+4\alpha}(\alpha+(t-2H_{0}t))}{C_{3}^{2}}\,,$
(67) $\displaystyle{\Gamma^{0}}_{22}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-1+4\alpha}(\alpha+(t-2H_{0}t))}{C_{3}^{2}}\,,$
(68) $\displaystyle{\Gamma^{0}}_{33}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-1+4\alpha}(\alpha+(t-2H_{0}t))}{C_{3}^{2}}\,,$
(69) $\displaystyle{\Gamma^{1}}_{10}$
$\displaystyle=\frac{3(\alpha+(t-2H_{0}t))}{t}\,,$ (70)
$\displaystyle{\Gamma^{2}}_{20}$
$\displaystyle=\frac{3(\alpha+(t-2H_{0}t))}{t}\,,$ (71)
$\displaystyle{\Gamma^{3}}_{30}$
$\displaystyle=\frac{3(\alpha+(t-2H_{0}t))}{t}\,,$ (72)

where, as usual, the Christoffel symbols are symmetric in the lower indices.
It follows that the corresponding Riemann tensor has the following non-
vanishing components

$\displaystyle{R^{0}}_{110}$
$\displaystyle=-\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+2\alpha)+5(1-2H_{0})t\alpha+3(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,,$
(73) $\displaystyle{R^{0}}_{220}$
$\displaystyle=-\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+2\alpha)+5(1-2H_{0})t\alpha+3(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,,$
(74) $\displaystyle{R^{0}}_{330}$
$\displaystyle=-\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+2\alpha)+5(1-2H_{0})t\alpha+3(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,,$
(75) $\displaystyle{R^{1}}_{010}$
$\displaystyle=\frac{3\left(\alpha-2\alpha^{2}+5(-1+2H_{0})t\alpha-3(1-2H_{0})^{2}t^{2}\right)}{t^{2}}\,,$
(76) $\displaystyle{R^{1}}_{221}$
$\displaystyle=-\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,,$
(77) $\displaystyle{R^{1}}_{331}$
$\displaystyle=-\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,,$
(78) $\displaystyle{R^{2}}_{020}$
$\displaystyle=\frac{3\left(\alpha-2\alpha^{2}+5(-1+2H_{0})t\alpha-3(1-2H_{0})^{2}t^{2}\right)}{t^{2}}\,,$
(79) $\displaystyle{R^{2}}_{121}$
$\displaystyle=\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,,$
(80) $\displaystyle{R^{2}}_{332}$
$\displaystyle=-\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,,$
(81) $\displaystyle{R^{3}}_{030}$
$\displaystyle=\frac{3\left(\alpha-2\alpha^{2}+5(-1+2H_{0})t\alpha-3(1-2H_{0})^{2}t^{2}\right)}{t^{2}}\,,$
(82) $\displaystyle{R^{3}}_{131}$
$\displaystyle=\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,,$
(83) $\displaystyle{R^{3}}_{232}$
$\displaystyle=\frac{9e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}(\alpha+(t-2H_{0}t))^{2}}{C^{2}_{3}}\,.$
(84)

From these, we obtain the following of the Ricci tensor

$\displaystyle R_{00}$
$\displaystyle=\frac{9\left(\alpha-2\alpha^{2}+5(-1+2H_{0})t\alpha-3(1-2H_{0})^{2}t^{2}\right)}{t^{2}}\,,$
(85) $\displaystyle R_{11}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+8\alpha)+17(1-2H_{0})t\alpha+9(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,,$
(86) $\displaystyle R_{22}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+8\alpha)+17(1-2H_{0})t\alpha+9(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,,$
(87) $\displaystyle R_{33}$
$\displaystyle=\frac{3e^{6\left(t-2H_{0}t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{-2+4\alpha}\left(\alpha(-1+8\alpha)+17(1-2H_{0})t\alpha+9(1-2H_{0})^{2}t^{2}\right)}{C^{2}_{3}}\,.$
(88)

The non-diagonal components vanish $R_{\mu\nu}=0$, for $\mu\neq\nu$. By
contracting with the inverse metric of (65) we obtain the following Ricci
scalar

$R=(1+\alpha)^{-2}\left[18t^{-2(1+\alpha)}\left(\alpha(-1+5\alpha)+11(1-2H_{0})t\alpha+6(1-2H_{0})^{2}t^{2}\right)\right]\,.$
(89)

The geodesic equations form the following system of non-linear differential
equations

$\displaystyle\frac{du^{0}}{ds}$
$\displaystyle=\frac{e^{-12H_{0}t}\left[3e^{6\left(t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}(-1+2H_{0})t^{1+4\alpha}\left((u^{1})^{2}+(u^{2})^{2}+(u^{3})^{2}\right)\right.}{C^{2}_{3}\,t}$
$\displaystyle-\frac{\alpha\left.\left(C^{2}_{3}e^{12H_{0}t}(u^{0})^{2}+3e^{6\left(t+\frac{\alpha}{H_{0}t}\right)}e^{2H_{0}t}t^{4\alpha}\left((u^{1})^{2}+(u^{2})^{2}+(u^{3})^{2}\right)\right)\right]}{C^{2}_{3}\,t}\,,$
(90) $\displaystyle\frac{du^{1}}{ds}$
$\displaystyle=-\frac{6(\alpha+(t-2H_{0}t))u^{0}u^{1}}{t}\,,$ (91)
$\displaystyle\frac{du^{2}}{ds}$
$\displaystyle=-\frac{6(\alpha+(t-2H_{0}t))u^{0}u^{2}}{t}\,,$ (92)
$\displaystyle\frac{du^{3}}{ds}$
$\displaystyle=-\frac{6(\alpha+(t-2H_{0}t))u^{0}u^{3}}{t}\,.$ (93)

Note that the geometries described in this section have a fractional character
manifest by the presence of the fractionality parameter $\alpha$. One can
easily see that at extreme fractionality $\alpha=0$ the Christoffel symbols
take the following form

$\displaystyle{\Gamma^{0}}_{00}$ $\displaystyle=0\,,$ (94)
$\displaystyle{\Gamma^{0}}_{11}$
$\displaystyle={\Gamma^{0}}_{22}={\Gamma^{0}}_{33}=\frac{3e^{t\left(6-10H_{0}\right)}(1-2H_{0})}{C_{3}^{2}}\,,$
(95) $\displaystyle{\Gamma^{1}}_{10}$
$\displaystyle={\Gamma^{2}}_{20}={\Gamma^{3}}_{30}=3-6H_{0}\,.$ (96)

For $\alpha=0$, the Riemann tensor takes the simpler form

$\displaystyle{R^{0}}_{110}$
$\displaystyle={R^{0}}_{220}={R^{0}}_{330}=-\frac{9(1-2H_{0})^{2}e^{t\left(6-10H_{0}\right)}}{C^{2}_{3}}\,,$
(97) $\displaystyle{R^{1}}_{010}$
$\displaystyle={R^{2}}_{020}={R^{3}}_{030}=-9(1-2H_{0})^{2}\,,$ (98)
$\displaystyle{R^{1}}_{221}$
$\displaystyle={R^{1}}_{331}={R^{2}}_{332}=-\frac{9e^{t\left(6-10H_{0}\right)}(1-2H_{0})^{2}}{C^{2}_{3}}\,,$
(99) $\displaystyle{R^{2}}_{121}$
$\displaystyle={R^{3}}_{131}={R^{3}}_{232}=\frac{9e^{t\left(6-10H_{0}\right)}(1-2H_{0})^{2}}{C^{2}_{3}}\,.$
(100)

It follows that the Ricci tensor at extreme fractionality has the form

$R_{00}=-27(1-2H_{0})^{2}\,,\qquad
R_{11}=R_{22}=R_{33}=\frac{27e^{t(6-10H_{0})}(1-2H_{0})^{2}}{C^{2}_{3}}\,.$
(101)

Thus, we can see from the above relations, that the components of the
geometric tensors are either constant or depend exponentially on time at
$\alpha=0$. At different fractionalities, the geometric tensors are given by
the previous formulas.

##  5 Discussions

In this paper, we have propose a new model of non-local mimetic dark matter
based on the FALVA construction. The FALVA-FMDDM generalizes previous FALVA
models of gravity by including a dark matter sector. These models are in
generally characterized by fractional space-time geometry. In order to
understand better the space-time fractionality, we have discussed in detail
two particular cases of the general FALVA-FMDDM, with a non-fractional and a
power law fractional mimetic field, respectively. In this cases, we have
computed the relevant geometrical objects as given by the standard
differential geometry. The results show that the fractional space-time is not
a trivial generalization of the standard space-time.

While the scope of the present paper is to propose and discuss the basic
geometric properties of the FALVA-FMDDM, further studies of this model are
necessary in order to understand the effect of fractionality on the physical
properties of the dark matter field. The most interesting problems at this
moment are a deeper analysis of the fractional geometries, including the
stability of the fractional space-time and the solutions of the geodesic
equations and the study of the fractional mimetic dark matter in different
cosmological scenarios. We hope to report on these topics in the future.

## Acknowledgements

We acknowledge R. A. El-Nabulsi for very useful comments and discussions.

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# Ozy: A General Orchestration Container

Glenn Osborne  Catavolt Inc
Alpharetta, GA 30022, USA
Email<EMAIL_ADDRESS>Tim Weninger  Department of Computer
Science and Engineering
University of Notre Dame
Notre Dame, IN 46556
Email<EMAIL_ADDRESS>
###### Abstract

Service-Oriented Computing is a paradigm that uses services as building blocks
for building distributed applications. The primary motivation for
orchestrating services in the cloud used to be distributed business processes,
which drove the standardization of the Business Process Execution Language
(BPEL) and its central notion that a service is a business process. In recent
years, there has been a transition towards other motivations for orchestrating
services in the cloud, e.g., XaaS, RMAD. Although it is theoretically possible
to make all of those services into WSDL/SOAP services, it would be too
complicated and costly for industry adoption. Therefore, the central notion
that a service is a business process is too restrictive. Instead, we view a
service as a technology neutral, loosely coupled, location transparent
procedure. With these ideas in mind, we introduce a new approach to services
orchestration: Ozy, a general orchestration container. We define this new
approach in terms of existing technology, and we show that the Ozy container
relaxes many traditional constraints and allows for simpler, more feature-rich
applications.

###### Index Terms:

SOA, SOC, coordination, orchestration, service

##  I Introduction

Service-Oriented Computing (SOC) is a paradigm that uses services as building
blocks for building distributed applications. Services perform procedures,
which can be anything from simple information requests to complicated business
processes. The original focus of SOC was the automation of business processes
by coordinating the services of multiple business partners. In recent years,
however, the Web has evolved into an ecosystem of Everything-as-a-Service
(XaaS) [1]; consequently, services orchestration has moved beyond the need for
just coordinating business processes and has evolved into a more general
problem of programming-in-the-large.

The basic Service-Oriented Architecture (SOA) is a relationship involving
three kinds of participants: Service Provider, Service Client and Service
Registry. The Extended Service-Oriented Architecture (ESOA) extends the basic
SOA with three tiers that address overarching concerns. The middle tier is a
service composition layer that facilitates orchestrations over service
definitions and service implementations [2].

The standard language for service orchestration is the Business Process
Execution Language (WS-BPEL or BPEL), which is based on the specifications for
SOAP, WSDL and UDDI [3]. Its design is centered on the notion of the business
process being the glue between interacting services.

In recent years, RESTful (REpresentational State Transfer) services have
emerged as a lightweight alternative to SOAP services. Whereas SOAP services
are based on remote procedure calls, RESTful services build upon the HTTP
hypermedia standards of resources, resource identifiers and representations.
The key elements of RESTful services are technology neutral components,
connectors and resource identifiers [4].

The standard technique for composing RESTful Web services is the ”mashup”.
There are roughly three kinds of mashups: data-oriented, process-oriented and
consumer-oriented. Unlike data and consumer-oriented mashups, which permeate
the Web, process-oriented mashups are less frequent, but most typically
provide for the composition of business processes [5].

There are several forces driving the proliferation and heterogeneity of
service-oriented computing. The adoption of utility computing on a pay-as-you-
go basis has allowed enterprises to develop applications more easily, cost
effectively, and reliably than ever before [6]. New service platforms, such as
Rapid Mobile App Development (RMAD) [7], facilitate opportunistic software
development by business analysts. Finally, the need to modernize legacy
systems has led to the micro-container concept that uses services to provide
accessibility and increased scalability of older software architectures [8].

RMADSAPLegacy ERPData Extraction{external use only}MediaBig DataSocial
Networkingfor
BusinessMetadataPhoneTabletThingFileCustomDriverRESTJDBCBAPISOAPJDBCUI
ServicesCustom HTTP/JSON

Figure 1: Heterogeneous services on an enterprise system. The Ozy container
aims to elegantly orchestrate these diverse services.

The Web service concept has evolved and now means more than just a business
process. As a result, we believe that the industry can benefit from a more
general treatment of services orchestration. Therefore, in this paper we
introduce a new approach to services orchestration with Ozy, a general
orchestration container based on the Oz Computation Model (OCM) [9]. In
addition to the description provided in the present work we have released the
orchestration framework as an open source software project under the Affero
GPL at https://github.com/ozyio.

To illustrate this new approach, its benefits and its restrictions, we show
examples of an RMAD platform for both the development and deployment of mobile
apps. As illustrated in Fig. 1, large enterprises typically orchestrate
several kinds of services including:

  * •

Swagger/RESTful services

  * •

WSDL/SOAP service

  * •

Database services (NOSQL and JDBC)

  * •

File services for serving media data

  * •

Legacy services to access SAP (BAPI) or other older ERP services

  * •

Proprietary HTTP/JSON services to support rich user experiences [10]

In its original sense, a service is simply a technology neutral, loosely
coupled, location transparent procedure [2]. However, the prevailing
techniques for orchestrating services are lacking support for the
heterogeneity illustrated here. BPEL, for example, has demonstrated success in
orchestrating business processes, but its central notion that a service is a
business process is too restrictive. Furthermore, BPEL suffers from two strong
criticisms: (1) it lacks a formal specification, and (2) its language is based
on a cumbersome XML-based syntax [11]. RESTful orchestration techniques are
also too restrictive because they also focus on business processes [12].
Moreover, the RESTful orchestration techniques that do exist are immature [5].

In the present work we present Ozy, a general orchestration container. Ozy is
based on the Oz Computation Model (OCM); although the Oz language is best
known for teaching multi-paradigm programming practices [13], the underlying
theory and constructs of the OCM are especially useful to services
orchestration. Specifically, we exploit implicit synchronization, dataflow
variables [14], lazy execution, failed values and partial termination [9].
Furthermore, the expressibility of the Oz language allows us to use
programming techniques not typically found in orchestration languages, such as
declarative functional programming and pattern matching [15]. To the best of
our knowledge, Ozy is the only orchestration engine that supports both
implicit synchronization and persistent execution state. In addition, the Ozy
orchestration container also includes the following contributions:

  1. 1.

A new notion of partial activation as a dual to the existing notion of partial
termination

  2. 2.

A technology neutral orchestration architecture inspired by the elements of a
network-based architecture

  3. 3.

An open source orchestration container that implements the orchestration
architecture (2), implements an Oz language interpreter based on the OCM, and
exploits the notions of partial activation (1) and partial termination to
support a persistent execution state

This paper describes the Ozy framework with special attention paid to the
limitations of current approaches. After a section on related work and a brief
introduction to the Oz computation model, the Ozy architecture and container
systems are described. We finally provide two illustrative examples that
highlights the Ozy orchestration container in real world scenarios.

##  II Related Work

Service-Oriented Computing (SOC) is a paradigm that uses loosely-coupled
services as the fundamental elements for building applications. SOC relies on
the Services-Oriented Architecture (SOA) and the Extended Services-Oriented
Architecture (ESOA) to define standards for publishing, finding, binding and
composing services [2]. Service-Oriented Middleware (SOM) supports the
service-oriented interaction pattern with the provisioning of software
features for deploying, publishing, finding, and binding services at runtime
[16, 17].

Composing services is an exercise in programming-in-the-large. DeRemer and
Kron argue that programming-in-the-large is a distinct and different
intellectual activity from programming-in-the-small, and consequently, a
different ”module interconnect language” should be used instead of a common
programming language, such as one typically used to implement modules.

In the traditional view, there are three service composition models:
Choreography, Orchestration and Wiring. Orchestration is the most used
composition model, and the de facto standard is BPEL, an orchestration
language based on the base specifications of SOAP, WSDL and UDDI [3]. Although
BPEL is the industry standard, it suffers from two big criticisms: (1) it
lacks formalization and (2) is based on a cumbersome XML syntax [11].

SOCK and JOLIE are two more-recent contributions that address some of the
deficiencies of BPEL. SOCK (Service-Oriented Computing Kernel) formalizes the
basic mechanisms of services communications and composition, therefore
addressing the lack of formalism in BPEL [18]. JOLIE (Java Orchestration
Language Interpreter Engine) builds on SOCK to create an expressive language
and orchestration interpreter engine, therefore addressing the cumbersome XML-
based syntax of BPEL [19, 20].

The open source Apache ODE (Orchestration Director Engine) project is the
underlying execution engine for several BPEL implementations, such as JBoss
RiftSaw and WSO2 Business Process Server. Apache ODE relies on the Apache
JaCOb (Java Concurrent Objects) framework to implement two key features
required of BPEL processes: concurrency and persistence of execution state.

In recent years, and because of the success of the RESTful architecture, a Web
service is more generally defined as a SOAP-based or RESTful-based service
that can be described, advertised and discovered through Internet-based
protocols [21]. SOAP services are based on remote procedure calls. In
contrast, RESTful services build on the HTTP hypermedia standards of
resources, resource identifiers and representations. The key elements of
RESTful services are technology neutral components, connectors and resource
identifiers [4].

Lightweight RESTful services are designed to ease consumption, composition and
have been successfully used to build community-driven services known as
mashups [22, 5]. Another approach for composing RESTful services is a
technique that uses BPEL to orchestrate RESTful services by wrapping RESTful
services with WSDL definitions [12].

Cloud computing has emerged in conjunction with Web services as a dominant
utility computing solution. Cloud vendors offer platform-as-a-service (PaaS)
and software-as-a-service (SaaS) applications on a pay-as-you-go basis
allowing enterprises to develop applications more easily, cost effectively,
and reliably than ever before [6]. Cloud services have become so prolific that
we now refer to the provider ecosystem as Everything-as-a-Service (XaaS) [1].

Opportunistic software methods are being used in cloud computing to drive down
the backlog of applications by allowing non-traditional software developers to
combine two or more software systems to create functionality that mainstream
software does not provide [23]. For example, Gartner recently identified the
Rapid Mobile App Development (RMAD) category of development tools [7]. RMAD is
opportunistic software development for mobile apps that offers simpler, faster
development by business analysts.

The need to reduce application backlogs is just one of many motivating forces
affecting service-oriented computing in the cloud. Service-oriented computing
has focused primarily on easy integration at the server-side at the price of
lower flexibility and ease-of-use at the client-side [24]. Focus is shifting
to client-side interactions with services that sometimes require innovative
service protocols [10]. Another force affecting cloud computing is the need to
modernize legacy software. The micro-container concept uses services to
provide accessibility and increased scalability of older software
architectures [8].

### The Missing Link

TABLE I: Feature comparison of existing orchestration languages: BPEL, JOLIE and the present work Ozy. Oz, itself, is not an orchestration language, but is included here for comparison. | BPEL | JOLIE | Oz | Ozy
---|---|---|---|---
Formal Specification | - | + | + | +
High-Level Language | - | + | + | +
Interpreted | + | + | - | +
Functional Programming | - | - | + | +
Long Running | + | + | - | +
Declarative Concurrency | + | + | + | +
Implicit Synchronization | - | - | + | +
Pattern Matching | - | - | + | +
Unification | - | - | + | +
Entailment | - | - | + | +

Composing services, and orchestrating in particular, has moved beyond the need
for coordinating business processes and has become a more general programming-
in-the-large problem. Although the Oz programming language is best known for
its support of multi-paradigm programming [13], the theory and constructs of
the Oz Computation Model (OCM) are especially useful for services
orchestration. Most notable are implicit synchronization, dataflow variables
[14], lazy execution, failed values and partial termination [9]. Furthermore,
the expressibility of the Oz language enables programming techniques not
typically found in programming-in-the-large [15]. In the present work, we
adapt the Oz Computational Model for practical use in service orchestration.
We call this adaption Ozy, and show that it is a more elegant, feature rich
orchestration container.

##  III The Oz Computation Model

The Oz Computation Model (OCM) [9] contains abstractions especially useful for
services orchestration. Specifically, we exploit implicit synchronization,
dataflow variables [14], lazy execution, failed values and partial termination
[9] provided by the OCM. Furthermore, the expressibility of the Oz language
allows advanced programming techniques, such as declarative functional
programming and pattern matching [15].

The dataflow variable is the most powerful OCM abstraction because it greatly
simplifies concurrency by hiding explicit synchronization, thereby giving us
implicit synchronization. Dataflow variables provide implicit synchronization
at the most atomic level–a single variable. A dataflow variable is created and
bound in separate steps. The single rule of accessing a dataflow variable is
very simple: execution waits until the dataflow variable is bound. Once bound,
execution continues. This property allows advanced programming techniques
never before seen in an orchestration language. The power of dataflow
variables is best explained by Smolka:

A basic problem with existing programming languages is that they delegate the
creation and coordination of concurrent computational activities to the
underlying operating system and network protocols. This has the severe
disadvantage that the data abstractions of the programming language cannot be
shared between communicating computational agents. Thus the benefits of
existing programming languages do not extend to the central concerns of
concurrent and distributed software systems [14] (emphasis added).

The present work, Ozy, uses the dataflow variable abstraction thereby allowing
service messages to be shared among interacting services independent of the
underlying operating system and network protocols.

For the following examples on dataflow variables, the complete Oz syntax can
be found in [9]. Our simple example, illustrated in Fig. 2, executes 3
concurrent threads. The third thread waits on variables $X$ and $Y$ before
continuing. This is an example of implicit synchronization because the third
thread is void of any syntax that explicitly waits for variables $X$ and $Y$
to be bound.

⬇

1thread X = 5 end

2thread Y = 7 end

3thread Z = X + Y end

Figure 2: Simple example of implicit synchronization with 3 threads. Despite
the lack of explicit syntax, implicit synchronization via dataflow variables
requires that the third thread waits for variables $X$ and $Y$ to be bound
(i.e., thread 1 and 2 to complete) before continuing.

A more sophisticated example of dataflow programming illustrated in Fig. 3
involves nested data structures and function calls111In Oz, function
signatures and function applications are delimited with curly braces..
Dataflow variables can be arbitrarily nested in both data structures and
logic. Because synchronization is implicit, logic can be written very cleanly,
void of explicit techniques that synchronize concurrency.

⬇

1fun {GetCustomerInfo Id}

2 fun {GetAverageSale Id} Answer in

3 thread

4 {Sleep 3 days}

5 Answer = 12345.67

6 end

7 return Answer

8 end

9 fun {GetLastSale Id}

10 123

11 end

12 customer(key:Id averageSale:{GetAverageSale Id} lastSale:{GetLastSale Id})

13end

14local C D in

15 C = {GetCustomerInfo ’acme’}

16 case C of customer(key:Id averageSale:AS lastSale:LS)

17 D = if LS > AS then ’Y’ else ’N’ end

18 end

19end

123456 Figure 3: Sophisticated example of dataflow programming that computes a
discount $D$ based on a complex, concurrent and implicitly synchronized work
flow. Existing orchestration languages are ill-equipped to elegantly
orchestrate this process.

Although our sophisticated example appears intuitive, there are actually deep
dependencies on concurrency and synchronization: (1) two numeric values are
compared that (2) depend on the result of pattern matching that (3) depends on
the result of a function call to GetCustomerInfo that constructs a compound
value that depends on two other nested function calls to GetAverageSale and
GetLastSale where (4) one of the nested function calls depends on (5) an
answer that (6) is fulfilled concurrently and takes three days to complete.
Dataflow variables greatly simplify concurrent programming. Without dataflow
variables, this example written in a mainstream programming language would
require explicit synchronization or a concurrency library, such as monadic
combinators for a functional language or an actor library for an imperative
language.

Another key OCM abstraction is the Kernel Language, a concise set of values
and statements that serve as the basic computational elements and as building
blocks for higher level computing techniques. The OCM defines formal
operational semantics and an abstract machine that specify exactly how to
execute a kernel language program. Informally, the abstract machine is a
process consisting of several semantic stacks and a shared store of dataflow
variables. Each semantic stack corresponds to a thread, and all semantic
stacks access the same store. Execution begins with an initial process state
and proceeds as a sequence of state transitions where a semantic stack
(thread) is chosen and reduced. Processing continues until all semantic stacks
are null. Semantic stacks can be reduced in parallel with synchronization on
the store of dataflow variables.

More formally, the Oz computation model is based on the following definitions:

  * $\sigma$ = single-assignment store (dataflow variables)

  * $s$ = kernel statement

  * $E$ = environment

  * $(s,E)$ = semantic statement

  * $[(s,E)]$ = semantic stack (ST)

  * $\\{[(s,E)]\\}$ = multiset semantic stack (MST)

A computation step begins by choosing from the multiset one runnable
(reducable) semantic stack giving us the following definition:

  * (a)

MST = {ST} $\uplus$ MST′

With a semantic stack chosen, a computation step is completed with the
following definition:

  * (b)

$(\\{\textrm{ST}\\}\uplus\textrm{MST}^{\prime},\sigma)\rightarrow(\\{\textrm{ST}^{\prime}\\}\uplus\textrm{MST}^{\prime},\sigma^{\prime})$

Program execution repeats steps (a) and (b) until termination [9].

The OCM executes over dataflow variables, which leads to the notion of partial
termination, a concept useful for reasoning about process state. To explain
partial termination, let us define a list as a pair (head, tail), where head
is any value and tail is a list value or null. An input stream is simply a
list with an unbound tail. If processing an input stream with an unbound tail,
the program may never terminate. However, if the input stream stops growing,
then based on the single rule for a dataflow variable (see above), the program
will eventually stop executing while it waits for the input stream to grow
(i.e., for a new tail to be bound). This is an example of partial termination.
It has not terminated completely because if the input stream grows, the
program will execute further, up to the next partial termination. If there are
no further input, then the program will execute no further. Partial
termination is particularly useful because it allows us to determine that a
long running process is waiting on an external message.

The new notion of partial activation occurs when a new semantic stack is added
to a partially terminated process, therefore guaranteeing at least one
semantic reduction. The process is then partially activated, which means that
it will execute, but it may or may not execute to complete termination as
described above. Generalizing partial activation allows for a variety of
activation scenarios in addition to the scenario above that uses a stream.
Partial activation combined with partial termination allows us to define the
life-cycle model for long-running processes. The life-cycle for long-running
processes behaves as follows:

  1. 1.

The process is created by initializing an abstract machine with an initial
semantic stack that is reducible. The process is now partially activated and
execution begins.

  2. 2.

The process waits on an external message and cannot execute further because
all semantic stacks are waiting on dataflow variables and therefore cannot be
reduced. The process is now partially terminated and eligible to be saved to
disk.

  3. 3.

An external message arrives for a partially terminated process. The process is
restored, from disk if necessary, and a new semantic stack is added to the
process. The process is now partially activated and execution begins.

  4. 4.

The process has been fully reduced, all semantic stacks are empty. The process
is now completely terminated.

Lazy execution and failed values are two other OCM constructs important to
services orchestration. Lazy execution allows for late binding of modules and
supports advanced functional programming techniques. Late binding is crucial
to large process configurations so that services are bound only when
necessary. Failed values are necessary to properly communicate failures. When
the computation of a dataflow variable fails, a failed value is stored in the
dataflow variable and the failure is propagated to other threads waiting on
that dataflow variable.

In summary, so far we have reviewed the Oz Kernel Language and Oz Abstract
Machine, as well as the highlights of the theory behind the Oz Computational
Model that are most relevant to services orchestration: implicit
synchronization, dataflow variables, lazy execution and failed values. These
constructs combined with the new notion of partial activation form the
technical foundation for the Oz language interpreter presented later in this
paper.

##  IV Ozy Architecture

The Ozy architecture is based on the general view of the network-based
architecture (net-arch) [4]. The conceptual elements of the net-arch are
Components, Connectors and Data. As illustrated in Fig. 4, components interact
by sending data through communications connectors (message passing).
Components, and only components, perform computation on state, which can be a
combination of data passed on connectors and private data known only to the
component. The connector is especially useful because it encapsulates the
technology used to communicate with other services. Components, connectors and
data messages are the basic abstractions used in the Ozy architecture.

ComponentComponentComponentConnectorData

Figure 4: Elements of the Network-Based Architecture (net-arch) wherein
components interact by sending data through communications connectors.

The manner in which Ozy employs the connector abstraction is what
differentiates it from SOA-based and RESTful-based architectures. The SOA
architecture and its BPEL are directly dependent and therefore constrained by
the underlying specifications for WSDL and UDDI. The RESTful architecture is
directly dependent on the uniform HTTP operations of GET, PUT, POST and
DELETE. In contrast, the Ozy architecture loosens the constraints on the
connector abstraction. A connector simply transfers data messages from one
interface to another without any semantic transformation. Internally, the
connector may be a subsystem that transforms and adapts, but when the message
is delivered, it is semantically unchanged.

We must emphasize that the Ozy architecture is designed for hiding services
implementations. Ozy does not define a new kind of services interface. It is
very important that public interfaces for composite services be founded on
industry standards, which is currently either SOAP or RESTful style
interfaces. By moving the dependencies on network interfaces into connector
abstractions (adapters), Ozy allows services containers to operate freely and
evolve independently as long as service invocations can be adapted between
private and public interfaces. Furthermore, private-only compositions can be
optimized to simply use local interfaces.

Service (Tenant)SOAP, REST, Database Driver, OtherComponent
InterfacesService(Process)Service(Process)Private StatePrivate
StateComponentDataConnectorrecordfunctorsprocedureOzy Container

Figure 5: The Ozy Architecture. Ozy does not define a new services interface;
industry standard SOAP or RESTful interfaces should still be employed, but the
Ozy Architecture does provide a better separation of concern between interface
and implementation.

Ozy exploits the conceptual elements of the net-arch to create a better
separation of concern between service interfaces and service implementations.
As shown in Fig. 5, an Ozy interface can be SOAP or RESTful as long as the
supporting connectors are available to adapt SOAP or RESTful messages to Oz
language elements, such as functors, records and procedures. Likewise, the Ozy
implementation can invoke any service; public or private, local or remote,
given that the supporting connectors can translate Oz language elements into
target interfaces. Finally, non-standard connectors can be installed to
transparently support interactions with non-standard protocols, such as NoSQL
drivers and legacy ERP interfaces.

##  V Ozy Container

The Ozy container is implemented in accordance with the Ozy architecture. The
container hosts two fundamental activities: message routing and computation.
The connector abstraction is realized by the computations that are performed
along the message paths. When a message reaches a process interface, Ozy
invokes an Oz language interpreter to compute the message.

Ozy employs an actor model for non-blocking asynchronous message routing.
There are two types of messages: Tell and Ask. Unlike Tell, the Ask message
requires a response and is built on an asynchronous Future/Promise library
that routes responses back to the requester.

Message InMessage OutRTTTPPPPPPThThThThThRRootTTenantPProcessThThreadPartially
TerminatedPartially ActivatedConnectorMessage

Figure 6: Message routing in Ozy. Message routing is based on the Ozy
Architecture where the net-arch components are Root, Tenant, Process and
Thread.

Message routing is intrinsically multi-tenant. Figure 6 illustrates the
message routing in Ozy, where the tenant and process services from Fig. 5
corresponds to the tenants and processes in Fig 6. Message routing is based on
the following hierarchy template:
/root/tenants/{tenantId}/processes/{processId} where curly braces denote
identifying parameters. Messages are first received in the root actor mailbox
where they are subsequently processed and routed to the appropriate tenant
actor. The tenant actor receives request messages in its mailbox where they
are subsequently processed and routed to process actors. If necessary, a
tenant actor will activate a process actor prior to sending it a message if
that process has been partially terminated and had its execution state
persisted to disk.

Messages received from external services must be correlated to process
instances and procedure invocations. A persistent correlation table is used to
map business attributes to process identifiers. For example, suppose that a
request to create a supply-order is being processed and a correlation entry is
made that associates the supply-order number with the running process id. Now
let us suppose the process partially terminates because the supply-order must
be approved before it can be fulfilled. Later, when the supply manager
approves (or disapproves) the supply-order we can use the correlation table to
find the partially terminated process, activate it and allow it to continue
executing.

⬇

1proc next_process_message(p, m)

2 (s, E) = correlate_with_semantic_stack(p, m)

3 p.MST = [(s, E)] $\uplus$ p.MST

4 p.run()

5end

Figure 7: Next Process Message. A message received by a Process must be
correlated to a semantic statement. A typical correlation simply invokes a
procedure.

Similarly, correlation entries must be made to map external attribute names to
dataflow variables. Figure 7 shows how an incoming process message must be
correlated to a semantic stack. Suppose that approving a supply-order involved
setting an approved-flag to true or false. Now suppose that the
createSupplyOrder process is partially terminated and waiting on the approved-
flag dataflow variable known internally, and that a correlation entry exists
with this fact. Later, when the supply manager approves (or disapproves) the
supply-order, the approved-flag correlation entry can be used to construct a
semantic stack that will set the approved-flag dataflow variable and allow the
process to continue executing.

createThread(stmt)saveProcessTo(stream)restoreProcessFrom(stream)These
components comprise aninterpreter that implementsthe Oz Computation
Model.partiallyTerminated()startThread(stmt,
env)runTimeSlice(duration)runTimeSlideEnded()Ozy ContainerPThThDataflow
MemoryStackStackConnectorMessagePrivate Access

Figure 8: Ozy Computation. If we zoom-in on Process and Thread from Fig. 6, we
see the intersection of message routing and computation. Process and Thread
implement the Oz Computation Model.

Figure 8 shows the intersection of message routing and computation. After a
message is correlated and translated into a semantic stack, semantic
computation can begin. The first step of computation is performed when Ozy
selects a time slice message for execution, thereby satisfying the OCM
definition of thread selection: MST = {ST} $\uplus$ MST′. The computation step
is completed when a thread actor runs a time slice by popping its semantic
stack and performing a semantic operation as a function of its environment and
the single-assignment store (dataflow memory), thereby satisfying the OCM
definition of a full computation step: ({ST} $\uplus$ MST′, $\sigma$)
$\rightarrow$ ({ST′} $\uplus$ MST′, $\sigma^{\prime}$).

##  VI Case Study

In this section we present two scenarios programmed with Ozy, which highlight
the limitations of current approaches.

⬇

1proc {GetPrice Quantity ClientLoc Product ?Euro} SupId in

2 {Orch.updateCSet cset(product:Product quantity:Quantity
clientLoc:ClientLoc)}

3 SupId = {Reg.getIdByQuery Product}

4 (SupLoc1 SupLoc2) = {Reg.getData SupId}

5 MyLoc = ’socket://localhost:2564’

6 Euro = {Supp.getEuro SupLoc1 Product} * Quantity

7 thread {Sleep 3000 millis} BuyTimeout=true end

8 case {WaitTwo BuyOk BuyTimeout}

9 of 1 then skip

10 [] 2 then {Quantity ClientLoc Product ’No’}

11 end

12end

13proc {Buy Quantity ClientLoc Product Conf} IdOrder BkId in

14 BuyOk = true

15 if Conf = ’Yes’ then

16 IdOrder = {Supp.order SupLoc2 Quantity ClientLoc Product}

17 BkId = {Bank.pay Reg.bankLoc ClientLoc SupLoc2 MyLoc Euro}

18 {Orch.updateCSet cset(bkId:BkId)}

19 {Receipt BkId IdOrder}

20 {Orch.commit ClientLoc}

21 end

22end

Figure 9: Typical Scenario. A typical business process that establishes a
correlation set, finds services in the registry, and coordinates interactions
with other business processes.

Our first scenario in Fig. 9 comes from the original JOLIE paper [20]. This
scenario is a typical business process involving five participants: a
customer, a market, a service register, a supplier and a bank. Like JOLIE,
this example demonstrates an approach more expressive and more programmer-
friendly than using BPEL XML; however, unlike JOLIE, this example uses
implicit synchronization on dataflow variables and pattern matching, which are
not available in JOLIE, BPEL or other orchestration languages. Specifically,
line 8 shows a pattern match implicitly synchronized on BuyOk and BuyTimeout.
The pattern match waits on a nondeterministic choice from the WaitTwo
procedure call. Computation is suspended at line 8 until either BuyTimeout is
bound at line 7 or BuyOk is bound at line 14.

⬇

1proc {Subscribe ClientLoc TankId ?Stream} Tail in

2 Stream = 0|Tail

3 {Orch.updateCSet cset(clientLoc:ClientLoc

4 tankId:TankId stream:Stream}

5 {PushNextEvent TankId Stream}

6end

7proc {PushNextEvent TankId Stream} in

8 thread {Sleep 1 minute} Proceed=true end

9 case {WaitTwo Stop Proceed}

10 of 1 then skip

11 [] 2 then

12 WaterLevel = {GetWaterLevel TankId}

13 if {WaterLevelChanged WaterLevel 0.05} Next in

14 case Stream of Head|Tail then

15 Tail = WaterLevel|Next end

16 end

17 {PushNextEvent TankId Stream}

18 end

19end

20proc {Unsubscribe TankId Stream}

21 Stop=true

22end

Figure 10: Non-typical Scenario. A non-typical Internet-of-Things scenario
that emits an endless stream of integers to a waiting device until it
unsubscribes.

Our second scenario in Fig. 10 is not a business process. Instead, it involves
an Internet-of-Things device that monitors water level in a water tank. The
service tracks the last water level reported and sends an update when the
level changes by more than 5%. Water levels are gathered on the local file
system as comma-delimited text records. Line 2 initializes the Stream dataflow
variable to a 0 water level value and an unbound tail. Line 3 establishes a
correlation set with the Stream dataflow variable, which will be monitored by
our connector. The procedure at Line 7 is called recursively (tail call
optimized) from Line 17 to push water level events. Every time the Stream is
extended, our connector is notified and transmits an integer to the device.
This example demonstrates advanced programming features and flexibility not
found in JOLIE, BPEL or other orchestration languages. Specifically, it uses a
device connector based on a WebSocket where the connection is first
established by the server for better security. Communication is a non-typical
bi-directional connection that streams integers to the device only when needed
at line 15, therefore avoiding inefficiencies and client-side polling. The
service emits an endless stream of integers through the dataflow variable at
line 15 until the connector unsubscribes at line 20.

##  VII Conclusions

Although BPEL XML and JOLIE have been used extensively to orchestrate business
processes, they have major limitations that are succinctly yet powerfully
addressed by applying lessons and technologies from the Oz computation model,
most importantly implicit synchronization via dataflow variables and pattern
matching. In this paper we present a description of the orchestration
framework, Ozy, and provide its implementation as an open source project. We
further show that the Ozy framework is easily used to program and perform
standard business processes, and we also demonstrate the flexibility of Ozy
with a long-running, implicitly synchronized, Internet-of-Things scenario.

This contribution opens the door for many possible follow on studies. One
avenue for future work is in mobile devices and the Internet-of-Things, which
are energy constrained. Ozy can be used for computational offloading, thereby
reducing end-to-end latency between a device and its cloud services, which
could be shown to improve the user experience and extend battery life.

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