[70] Dolan SL, Garcia S. Managing by values: Cultural redesign for strategic organizational change at the dawn of the twenty-first century. The Journal of Management Development.

[71] İzci F, Arslan NT. Bilgi toplumuna geçiş sürecinde örgütsel yapıda meydana gelen değişimler: Bilişim teknolojisi örgütsel yapı ilişkisi. İÜ Siyasal Bilgiler Fakültesi Dergisi.

[72] Mintzberg H. The Structuring of Organizational Cultures. Upper Saddle

Press; 1993

**138**

[83] Şişman M. Örgüt kültürü, No:732. Eskişehir: Anadolu Üniversitesi Yayınları; 1994

[84] Schein EH. Organizational Culture and Leadership. San Francisco: Josey-Bass Inc. Publishers; 1992

[85] Denison DR, Neale W. Denison organizational culture survey. Ann Arbor, MI: Denison Consulting; 2000

### *Edited by Süleyman Davut Göker*

Values, attitudes, and behaviors constitute an organization's culture and employees both share and use them on a daily basis in their work. This book aims to briefly portray a new interpretation of organizational culture varying from the profusion of literature in the following ways: it attempts to include how cultures are created organically or through consistent planning and action in different organizations such as education, business, and health; focusing more on change, innovation, and learning opportunities. It also aims to provide leaders with experiences and reflections on how to initiate an organizational culture change. Finally, this book is expected to extend new perspectives and practices for both potential and actual managers of organizations contributing to the current debate on how to transform organizations into innovative and learning cultures.

Published in London, UK © 2021 IntechOpen © Anest / iStock

A Closer Look at Organizational Culture in Action

A Closer Look

at Organizational Culture

in Action

*Edited by Süleyman Davut Göker*


*4.6. Exogenous ABA Treatment*

A seed germination assay was performed as described above. For the ABA sensitivity assay, after-ripened seeds of NIL-NIP and NIL-ZS97 were treated with a series of ABA solutions (0, 1, 5, 10, 20, 50, and 100 μM). Dimethyl sulfoxide (DMSO) was used to dissolve ABA.

Subsequently, a 20 μM portion of ABA was chosen to treat after-ripened NIL-NIP and NIL-ZS97 to further measure the candidate gene expression. DMSO was used as control (CK). Embryos germinated for three days were used for RNA isolation.


4. Conclusions

This chapter investigated a piecewise parallel onboard optimal control algorithm to solve the optimal control issues in complex nonlinear dynamic systems in aerospace engineering. To test the validity of the proposed two-phase optimal control scheme, the long-term tether libration stability and fast nano-satellite deorbit under complex environmental perturbations and the libration suppression for PSE system are considered. For EDT system, instead of optimizing the control of fast and stable nano-satellite deorbit over the complete process, the current approach divides the deorbit process into a set of intervals. For the PSE system, each time interval is set depends on the minimize transfer time for equal transfer length interval. Within each interval, the predicting phase simplifies significantly the optimal control problem. The dynamic equations of libration motion are further simplified to reduce computational loads using the simple dynamic models. The trajectory of the stable libration states and current control input is then optimized for the fast deorbit within the interval based on the simplified dynamic equations. The tracking optimizes the trajectory tracking control using the finite receding horizon control theory within the time interval corresponding to the open-loop control state trajectory with the same interval number. By applying the close-loop control modification, the system motions are integrated without any simplification of the dynamics or environmental perturbations and the instantaneous states of the orbital and libration motions. The i-th time interval's closed-loop tracking is processed in tracking phase while the

χ

(i + 1)-th time interval's optimal state trajectory is predicted in the predicting phase. This prediction is based on the error between the real state and the predicting state in the (i-1)-th time interval. By repeating the process, the optimal control problem can be achieved in a piecewise way with dramatically reduced computation effort. Compared with the current on–off control where the stable libration motion is the only control target, numerical results show that the proposed optimal control scheme works well in keeping the libration angles within an allowed limit.


**1. Introduction**

Discrete-time queues with batch service have numerous applications in various areas such as transportation systems, traffic, production, manufacturing, telecommunication, and cloud computing. In various real-life settings, it is often noted that the jobs are served in batches. The server may serve with fixed maximum or variable capacity in batch service systems. For more details on batch service queues, one may refer to Chaudhry and Templeton [1] as well as Medhi [2]. Discrete-time queues are more notable in systems' modelling, see [3–5].

In discrete-time queues, it is assumed that arrivals and departures occur at boundary epochs of time slots. Further, discrete-time queues deal with an early arrival system (EAS) or a late arrival system with delayed access (LAS-DA). For more on this, see Hunter [4]. We may note that EAS and LAS-DA policies are similar to departure-first (DF) and arrival-first (AF), respectively; see Gravey and Hébuterne [6].

Several researchers study single-server batch-service discrete-time queues with various phenomena, see [7–13]. In Gupta and Goswami [10], they discuss a discrete-time finitebuffer general bulk service queue under both LAS-DA and EAS policies. The model involving batch-size-dependent service in a discrete-time queue where inter-arrival times and the service times follow geometric and general distribution, respectively, has been discussed by Banerjee et al. [14]. In Yi et al. [13], the authors analyze a discrete-time Geo/*Ga*,*Y*/1/N queue, where service is initiated only when the number of jobs in the system is at least '*a*'. In Zeng and Xia [15], the authors discuss M/*Ga*,*b*/1/N queue where service is in batches with minimum threshold *a*, maximum capacity *b* and the buffer size, *N*, finite or infinite.

At some point, finding the roots of the characteristic equation seemed difficult, mainly when the number involved was large. Several researchers have made these comments. Given this, the procedure for solving queueing models led to the matrix-analytic or matrixgeometric method. In this connection, see the remark below. Following Chaudhry et al. [16],

**Citation:** Chaudhry, M.; Goswami, V. The *Geo*/*Ga*,*Y*/1/*N* Queue Revisited. *Mathematics* **2022**, *10*, 3142. https:// doi.org/10.3390/math10173142

Academic Editors: Gurami Tsitsiashvili and Alexander Bochkov

Received: 25 July 2022 Accepted: 23 August 2022 Published: 1 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

other researchers have attempted to show that roots can be found, and using them leads to nice analytic and numerical solutions. Gouweleeuw [17] shows that the roots' approach for finding probabilities from generating functions in precise expressions is effective. Further, in the case of large buffer size, solving simultaneous equations gives rise to poor reliability and takes considerable time; see Powell [18] (p. 141), who, while dealing with the model *M*/*Dc*/1, states that when *c* ≥ 40 the method using simultaneous equations breaks down leading to negative probabilities. The goal of this paper is to give an alternative solution that is analytically powerful, simple, and easy to implement numerically. It may be stated further that this method has not been used to discuss the discrete-time queueing system that deals with batch services and a finite buffer.

In real-world systems, we encounter many finite-buffer systems such as telecommunication networks. Because of this, we study the *Geo*/*Ga*,*Y*/1/*N* model under the assumption of late arrival and delayed access system (LAS-DA). Here, we assume that the single server with variable service capacity will process the jobs only when there are at least '*a*' jobs in the system. In Yi et al. [13], the authors found the queue-length distribution at post-departure by solving simultaneous equations and random epoch by applying the "rate in = rate out" arguments. We develop an alternate process to find the queue-length distributions at post-departure and random epochs.

The principal contributions of this work may be summed up as follows:

The remaining paper is structured as follows. Section 2 specifies the model. Section 3 analyzes the *Geo*/*Ga*,*Y*/1/*N* system and finds queue-length distributions for the LAS-DA policy. Section 4 examines various system performance measurements. Section 5 provides some numerical results and, finally, the paper is concluded in Section 6.

**Remark 1.** *It may be useful to comment on the matrix-analytic and the root-finding method. Kendall [19] made a famous remark that queueing theory wears the Laplacian curtain. Kleinrock [20] (p. 291) states, "One of the most difficult parts of this method of spectrum factorization is to solve for the roots". Neuts (see Neuts' book [21] and also Stidham [22]) states, "In discussing matrix-analytic solutions, I had pointed out that when the Rouch' roots coincide or are close together, the method of roots could be numerically inaccurate. When I finally got copies of Crommelin's papers, I was elated to read that, for the same reasons as I, he was concerned about the clustering of roots. In 1932, Crommelin knew; in 1980, many of my colleagues did not . . . ". Following this, several other*

*researchers made similar comments. Given this, the procedure for solving queueing models led to the matrix-analytic or matrix-geometric method. In response, Chaudhry et al. [16] showed that the roots can be found even when the number involved is large. (This was done when MATHEMATICA OR MAPLE failed to give those roots. We can now use this software to find roots.)*

## **2. Model Description**

We consider a *Geo*/*Ga*,*Y*/1/*N* queue where jobs arrive following a Bernoulli process with parameter *λ*. A single server processes the jobs on a first-come-first-served (FCFS) discipline in batches with a random capacity *Y* possessing a probability mass function (pmf) *P*(*Y* = *i*) = *yi*, *i* = *a*, *a* + 1, ... , *b* with ∑*bi*=*a yi* = 1, the probability generating function (PGF) *Y*(*z*) = ∑*bi*=*a yizi*, mean *E*(*Y*) = ∑*bi*=*a iyi* and *Y*-(*z*) = *zbY*(*z*−1). We assume the minimum and maximum threshold values of the random variable *Y* as *a* and *b*, respectively. When there are at least '*a*' jobs in the queue, the server commences serving a batch of size *i* with probability *yi* (when there are *Y* ≤ *b*, it takes all of them). When the number of jobs comes down below a threshold value *a*(≥ 1) in the system, the server remains idle but awaits the number of jobs to rise to *a*; when it attains *a*, it resumes service. The service times {*Sn*, *n* ≥ 1} are independently and identically distributed (iid) with arbitrary pmf *sk* = *P*(*Sn* = *k*), *k* = 1, 2, ... and *s*0 = 0, the PGF *S*(*z*) = ∑∞*i*=1 *sizi* and mean service time *E*(*S*) = *s* = *ddzS*(*z*)|*z*=1= 1/*μ*.

The processing times of the jobs are independent of the arrival process and the number of jobs served. The waiting buffer has a finite capacity *N* with *b* ≤ *N*. Thus, in the system, no more than *N* + *b* jobs can be available anytime. We presume offered load of the system as *ρ* = *λE*(*S*)/*E*(*Y*). In LAS-DA policy, arrivals occur in (*u*−, *u*), and departures take place in (*u*, *u*+); arrivals supersede departures. Figure 1 describes the various time periods at which events occur. For more details on this, one may refer to [5,6].

(u+,(u +

:Potential arrival epoch :Potential departure epoch

1)−):Outside observer's interval u+ :Epoch after a potential departure

∗:Outside observer's epoch u− :Epochprior toapotentialarrival

**Figure 1.** Various time epochs in LAS-DA.

## **3. Queue-Length Distributions**

Here, we find steady-state queue-length distributions at various epochs of *Geo*/*Ga*,*Y*/1/*N* queue with the LAS-DA policy.

#### *3.1. Post-Departure Epoch Probabilities*

Let *Q*+- be the number of jobs in the queue after completing the th service. Suppose *A*-+1 and *Y*-+1 represent the number of arrivals throughout the processing time on the (- + 1)th job and the processing capacity of (- + 1)th service, respectively. As per the batch service rule, the departure epoch queue lengthis

$$Q\_{\ell+1}^{+} = \min \left( \left( Q\_{\ell}^{+} - Y\_{\ell+1} \right)^{+} + A\_{\ell+1} N \right)\_{\prime},$$

where *x*+=max(*x*, 0). The probability distribution of *A*-+1 is

$$\begin{aligned} k\_n &= P(A\_{\ell+1} = n) = \sum\_{j=1}^{\infty} P(A\_{\ell+1} = n | S\_{\ell+1} = j) P(S\_{\ell+1} = j), \\ &= \sum\_{j=n}^{\infty} s\_j \binom{j}{n} \lambda^n (1 - \lambda)^{j-n}, \ n \ge 0. \end{aligned}$$

Here we may note that arrivals are generated by a Bernoulli sequence by the property of geometric interarrival times. In LAS-DA, if the service time of the (- + 1)th job is *j* slots, then there will be *j* time slots where *n* arrivals may occur. One may note that (*j n*)*λn*(1 − *λ*)*j*−*n* is the probability of *n* arrivals in *j* slots. Let *K*(*z*) = ∑ ∞ *n*=0 *knzn* be the probability generating function of the sequence {*kn*, *n* = 0, 1, . . .}. Thus

$$K(z) = \sum\_{n=0}^{\infty} \sum\_{j=n}^{\infty} s\_j \binom{j}{n} \lambda^n (1-\lambda)^{j-n} z^n = \sum\_{j=0}^{\infty} s\_j (1-\lambda+\lambda z)^j = S(1-\lambda+\lambda z).$$

Transition probabilities in one step of underlying Markov chain for *pij* = lim -→∞*Pr*{*Q*+ -+1 = *j*|*Q*+ - = *i*} are given as

$$p\_{ij} = \begin{cases} k\_{jr} & 0 \le i \le a, \ 0 \le j \le N - 1, \\ k\_j \sum\_{r=i}^b y\_r + \sum\_{r=a}^{i-1} y\_r k\_{j-i+r}, & a+1 \le i \le b-1, \ i-a-1 \le j \le N-1, \\ \sum\_{r=a}^b y\_r k\_{j-i+r}, & b \le i \le N, \ i-b \le j \le N-1, \\ \ell\_{N\_r} & 0 \le i \le a, \ j = N\_r \\ \ell\_N \sum\_{r=i}^b y\_r + \sum\_{r=a}^{i-1} y\_r \ell\_{N-i+r}, & a+1 \le i \le b-1, \ j = N\_r \\ \sum\_{r=a}^b y\_r \ell\_{N-i+r} & b \le i \le N, \ j = N, \end{cases} \tag{1}$$

where *j* = ∞ ∑ *r*=*j kr* and *kj* with *j* < 0 defined to be zero, and which leads to the transition probability matrix *P* = (*pij*) as

*P* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0 1 ... *j* ...  *N* 0 00 *b* ∑ *r*=*a yrkj*−*N*+*r* . . . *b* ∑ *r*=*a yrkr*−1 *b* ∑ *r*=*a yrr* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The (- + 1)th batch service starts when there are '*a*' jobs in the queue, and the state changeover occurs from 0 ≤ *i* ≤ *a* − 1 to 0 ≤ *j* ≤ *N* − 1. If the state changeover is from state *i* ≥ *a* to state 0 ≤ *j* ≤ *N* − 1, then there is a busy period between the leaving epoch of the th batch and the commencement of processing (- + 1)st batch. Let the steady-state probability *p*+ = {*P*+0 , *P*+1 , ... , *P*+*n* , ... , *P*+*N* } represent - jobs at departure epochs. Then, in steady-state, *p*+ = *p*+*P* can be expressed as follows:

$$\begin{aligned} P\_j^+ &= \sum\_{i=0}^a P\_i^+ k\_j + \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{r=i}^b y\_r k\_j + \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{r=a}^{i-1} y\_r k\_{j-i+r} + \sum\_{i=b}^N P\_i^+ \sum\_{r=a}^b y\_r k\_{j-i+r} \\ &0 \le j \le N - 1, \end{aligned} \tag{2}$$

$$P\_N^+ = \sum\_{i=0}^a P\_i^+ \ell\_N + \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{r=i}^b y\_r \ell\_N + \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{r=a}^{i-1} y\_r \ell\_{N-i+r} + \sum\_{i=b}^N P\_i^+ \sum\_{r=a}^b y\_r \ell\_{N-i+r}, \tag{3}$$

where the normalization condition is *N* ∑ *j*=0 *P*+*j* = 1. It may be noted that Equation (3) is redundant and will not be considered in analysis hereafter. Specify PGF of *P*+*j* as *P*+(*z*) = *N* ∑ *j*=0 *P*+*j zj*. Multiplying Equation (2) by *zj* and then adding overall *j*, we obtain

$$\begin{split} P^{+}(z) &= \sum\_{j=0}^{N-1} k\_{j} z^{j} \sum\_{i=0}^{a} P^{+}\_{i} + \sum\_{j=0}^{N-1} z^{j} \sum\_{i=a+1}^{b-1} P^{+}\_{i} \left( \sum\_{r=i}^{b} y\_{r} \right) k\_{j} \\ &+ \sum\_{j=0}^{N-1} z^{j} \sum\_{i=a+1}^{b-1} P^{+}\_{i} \sum\_{r=a}^{i-1} y\_{r} k\_{j-i+r} + \sum\_{j=0}^{N-1} z^{j} \sum\_{i=b}^{N} P^{+}\_{i} \sum\_{r=a}^{b} y\_{r} k\_{j-i+r} + P^{+}\_{N} z^{N} \\ &\quad P^{+}(z) \left[ 1 - K(z) Y \left( \frac{1}{z} \right) \right] = K(z) \sum\_{i=0}^{a-1} P^{+}\_{i} \left( 1 - z^{i} Y \left( \frac{1}{z} \right) \right) + P^{+}\_{N} z^{N} \\ &\quad + K(z) \sum\_{i=a}^{b} P^{+}\_{i} \left( \sum\_{r=i}^{b} y\_{r} - z^{i} \sum\_{r=i}^{b} z^{r} \right) - \sum\_{i=a}^{b} y\_{i} \sum\_{j=i+1}^{N} P^{+}\_{j} + \sum\_{r=N-j+a}^{\infty} k\_{r} z^{r+j-i} \\ &\quad - \left( \sum\_{i=0}^{a-1} p\_{i}^{+} + \sum\_{i=a}^{b} P^{+}\_{i} \sum\_{r=i}^{b} y\_{r} \right) \sum\_{j=N}^{\infty} k\_{j} z^{j} .\end{split}$$

Simplifying the above equation, we get

$$\begin{split} P^{+}(z) &= \frac{K(z)\left[\sum\_{i=0}^{a-1} P\_{i}^{+}\left(z^{b} - z^{i}Y'(z)\right) + \sum\_{i=a}^{b-1} P\_{i}^{+}\left(\sum\_{r=i}^{b} y\_{7}z^{b} - z^{i}\sum\_{r=i}^{b} y\_{7}z^{b-r}\right)\right]}{z^{b} - K(z)Y'(z)} \\ &+ \frac{z^{N+b}\left(P\_{N}^{+} - \sum\_{i=a}^{b} y\_{i}\sum\_{j=i+1}^{N} P\_{j}^{+}\sum\_{r=N-j+a}^{\infty} k\_{7}z^{r+j-i-N}\right)}{z^{b} - K(z)Y'(z)} \\ &- \frac{z^{N+b}\left(\sum\_{i=0}^{a-1} P\_{i}^{+} + \sum\_{i=a}^{b} P\_{i}^{+}\sum\_{r=i}^{b} y\_{r}\right)\sum\_{j=N}^{\infty} k\_{j}z^{j-N}}{z^{b} - K(z)Y'(z)}. \end{split} \tag{4}$$

Only the first expression on the right side of the Equation (4) will add to the coefficient of *zj*, *j* = 0, 1, ... *N*. To the right of Equation (4), we ignore the second and third expressions, which consist of an output of *z* higher than *N* + *b*. These are not required as we want

to compare the coefficients of *zj* for *j* ≤ *N* on both sides in Equation (4) to find *P*+*j* for *j* = 0, 1, . . . , *N*. Let

$$P\_N^+(z) = \frac{K(z)\left[\sum\_{i=0}^{a-1} P\_i^+\left(z^b - z^i Y'(z)\right) + \sum\_{i=a}^{b-1} P\_i^+\left(\sum\_{r=i}^b y\_r z^b - z^i \sum\_{r=i}^b y\_r z^{b-r}\right)\right]}{z^b - K(z)Y'(z)},\tag{5}$$

which is equivalent to the PGF of an infinite buffer case. The function *P*+*N* (*z*) is fully determined once *P*+*i* , *i* = 0, 1, ... , *b* − 1 are known. One may observe that when *ρ* < 1, Equation (5) denotes the PGF of discrete-time infinite buffer *Geo*/*Ga*,*Y*/1 queue. We can calculate the probabilities for *ρ* ≥ 1 in the case of a finite buffer *Geo*/*Ga*,*Y*/1/*N* queue.

**Remark 2.** *Using a* = 1*, the model is reduced to Geo*/*GY*/1/*N queue.*

**Remark 3.** *Taking y*1 = 1*, yi* = 0, ∀ 2 ≤ *i* ≤ *b and a* = 1*, the model becomes Geo*/*G*/1/*N queue and Equation* (5) *establishes PGF as P*+*N* (*z*) = *P*+0 *K*(*z*)(1−*z*) *K*(*z*)−*z , where K*(*z*) = *S*(1 − *λ* + *λz*)*, which corresponds to the results of [23].*

**Remark 4.** *Taking yb* = 1 *and yi* = 0, ∀ *i* = *b, the model is reduced to Geo*/*G*(*a*,*b*)/1/*N and Equation* (5) *establishes the PGF as P*+*N* (*z*) = *K*(*z*) *b*−1 ∑ *i*=0 *P*+*i* (*zb*−*zi*) *zb*−*K*(*z*) *.*

Intending to establish a unified method to solve the queueing system *Geo*/*Ga*,*Y*/1/*N*, we obtain {*P*+*n* }*N*0 from *P*+(*z*) by using the roots of characteristic equation and partialfraction expansion. The literature on queueing systems shows that arrival/service-time distributions that possess the generating function as a rational function deal with the broad class of distributions see [24]. For this, we suppose that *K*(*z*) = *S*(1 − *λ* + *λz*) as a rational function in *z*, specified by

$$K(z) = S(1 - \lambda + \lambda z) = \frac{f(z)}{g(z)},$$

where *f*(*z*) and *g*(*z*) are polynomials of degree *m* and *s*, respectively, where *m* and *s* can have any value, e.g., *m* can be greater than *s*, e.g., see Example 3(ii). Thus, we have from Equation (5),

$$P\_N^+(z) = \frac{f(z)\left[\sum\_{i=0}^{a-1} P\_i^+\left(z^b - z^i Y'(z)\right) + \sum\_{i=a}^{b-1} P\_i^+\left(\sum\_{r=i}^b y\_r z^b - z^i \sum\_{r=i}^b y\_r z^{b-r}\right)\right]}{z^b g(z) - f(z)Y'(z)}.\tag{6}$$

The denominator of Equation (6) is a polynomial of degree *b* + *s* which when equated to zero has *b* + *s* roots inside, on, or outside the unit circle | *z* |= 1, say, *γ*1 = 1, *γi* (*i* = 2, 3, . . . , *b* + *s*).

**Remark 5.** *If the denominator zbg*(*z*) − *f*(*z*)*Y*-(*z*) *of Equation* (6) = 0 *obtains roots close to each other or repeated roots, we may obtain them by applying advanced software packages, for instance, MATHEMATICA or MAPLE. The MAPLE script for calculating repeated roots is exemplified below for Equation*

$$
\mu(y) = (y-2)(y-5)^2(y-7)^3(y-11).
$$

*restart* : *Digits* := 10 : *with*(*RootFinding*) :

*m* := (*y* − 2)(*y* − 5)2(*y* − 7)3(*y* − 11);

*Analytic*( *u*, *y*, *re* = −1..10, *im* = −2 ..10 );

7.00000000000000, 7.00000000000000, 7.00000000000000, 5.00000000000000, 5.00000000000000, 2.00000000000000, 11.00000000000000

According to Rouché's theorem, the denominator of Equation (6) has *b* zeros say, *γi* (*i* = 1, 2, ... , *b*) inside the unit circle. As *P*+*N* (*z*) converges in | *z* |≤ 1, the *b* zeros within the unit circle of the denominator should cancel with the *b* zeros of the numerator. After canceling the *b* factors in the numerator and denominator, we can re-write Equation (6) as

$$P\_N^+(z) = T \left( \mathbb{C}(z) + \frac{f\_1(z)}{\prod\_{\substack{b+s\\i=b+1}}^{b+s} (z-\gamma\_i)} \right)' \tag{7}$$

where *C*(*z*) = *n*0 ∑ *i*=0 *cizi* and *T* is a normalizing constant. Note that when 2*m* < *s*, then *C*(*z*) will be zero. In the partial-fraction process, a slight modification is needed ([25], p. 221) when all the roots are not distinct. Since we are looking at the finite buffer queue system, three instances appear here.

The expression (7) is tractable for inversion. Applying partial-fraction expansion to Equation (7) yields

$$P\_N^+(z) = T\left(\mathbb{C}(z) + \sum\_{i=b+1}^{b+s} \frac{M\_i}{z - \gamma\_i}\right),\tag{8}$$

where

$$M\_i = \frac{f\_1(\gamma\_i)}{\prod\_{j=b+1, j \neq i}^{b+s} (\gamma\_i - \gamma\_j)}.$$

Using Equation (8), we have

$$P\_n^+ = \begin{cases} T\left(c\_n + \sum\_{i=b+1}^{b+s} \frac{-M\_i}{\gamma\_i^{n+1}}\right), & \text{if } n = 0, 1, \dots, n\_{0, \text{r}} \\\\ T\sum\_{i=b+1}^{b+s} \frac{-M\_i}{\gamma\_i^{n+1}}, & \text{if } n = n\_0 + 1, n\_0 + 2, \dots, N. \end{cases} \tag{9}$$

Employing the normalization condition *N* ∑ *n*=0 *P*+*n* = 1, we obtain the only unknown *T* as

$$T = \begin{cases} \left(\sum\_{i=0}^{n\_0} c\_i - \sum\_{i=b+1}^{b+s} \frac{M\_i}{\gamma\_i} \times \frac{1 - \gamma\_i^{-(N+1)}}{1 - \gamma\_i^{-1}}\right)^{-1}, & \text{if } \rho \neq 1, \\\\ \left(\sum\_{i=0}^{n\_0} c\_i - M\_{b+1}(N+1) - \sum\_{i=b+2}^{b+s} \frac{M\_i}{\Psi\_i} \times \frac{1 - \Psi\_i^{-(N+1)}}{1 - \Psi\_i^{-1}}\right)^{-1}, & \text{if } \rho = 1. \end{cases} \tag{10}$$

Thus, once all the roots are known, we can get the distribution for the number in queue.

**Remark 6.** *It may be noted that it is also possible to find the probabilities* {*P*+*n* }*N*0 *by assuming the solution of the form P*+*j* = *Czj, where C* = 0*. Unfortunately, if we use this method, we have to solve for N simultaneous equations.*

#### *3.2. Relationship between the Queue-Length Distributions at Post-Departure and Random Epochs*

This sub-section establishes associations between probability at random and postdeparture epochs by basic probabilistic reasoning and discrete-time renewal theory. In steady-state, let {*Pj*}*N*0 and {*P*−*j* }*N*0 be the probabilities representing the number of jobs in the queue at random times and before arrival, respectively. Since the inter-arrival times use geometric distribution, the arrivals are independent of other events. Thus, it implies that *Pj* = *P*−*j* , ∀ *j* = 0, 1, ... , *N*; for details, see Boxma and Groenendijk [27]. If the server is idle, there are < *a* jobs in the queue. Suppose *Nq* is the number of tasks in the queue at some random time. At a random epoch, the steady-state probabilities are *Pn*,0 = *P*(*Nq* = *n*, server idle), 0 ≤ *n* ≤ *a* − 1, and *Pn*,1 = *P*(*Nq* = *n*, server busy), 0 ≤ *n* ≤ *N*. Given this,

$$P\_j = \begin{cases} P\_{j,0} + P\_{j,1} & \text{if } 0 \le j \le a - 1 \\ P\_{j,1} & \text{if } a \le j \le N \end{cases}$$

Let the limiting pmf of the elapsed service time be *s*ˆ-, which is determined by *s* ˆ - = *μ* ∞ ∑ *r*=-+1 *sr*, - ≥ 0 (see, [5] (p. 20)), and ˆ *k*- be the probability that the number of arrivals during an elapsed service time is -. This yields

$$\pounds\_{\ell} = \sum\_{i=\ell}^{\infty} \binom{i}{\ell} \lambda^{\ell} (1-\lambda)^{i-\ell} \pounds\_{i\prime} \; \ell = 0, 1, \dots$$

If *E*∗ is the mean inter-departure time of processing batches, 1/*E*∗ represents the departure rate. At the batch departure epoch, if the number of jobs in the queue is less than *a*, the subsequent batch departure occurs after an idle time and the service time processing time. Otherwise, the release of the next batch takes place following the processing time of the subsequent batch. This gives,

$$E^\* = E(\mathcal{S}) \left( 1 - \sum\_{i=0}^{a-1} P\_i^+ \right) + \sum\_{i=0}^{a-1} P\_i^+ \left( \frac{a-i}{\lambda} + E(\mathcal{S}) \right) = E(\mathcal{S}) + \sum\_{i=0}^{a-1} P\_i^+ \frac{(a-i)}{\lambda} \cdot \mathcal{S}$$

**Remark 7.** *It can also be put down as E*∗ = *E*(*S*) + *a* ∑ *i*=1 *P*+*a*−*i iλ . When a* = 1*, it matches with Chaudhry and Chang [7].*

**Theorem 1.** *The random- and post-departure-epoch probabilities* {{*Pj*,0}*a*−1 0 , {*Pj*,1}*N*0 }*, and* {*P*+*j* }*N*0 *are related by*

$$P\_{j,0} = \frac{\sum\_{i=0}^{j} P\_i^+}{\sum\_{i=0}^{a-1} (a-i)P\_i^+ + \lambda E(S)}, \quad 0 \le j \le a-1,\tag{11}$$

$$\begin{split} P\_{j,1} &= \left(1 - \sum\_{i=0}^{a-1} P\_{i,0}\right) \cdot \left[\sum\_{i=0}^{a-1} P\_i^+ \hat{k}\_j + \sum\_{i=a}^{b-1} P\_i^+ \left(\sum\_{m=i}^b y\_m\right) \hat{k}\_j + \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{m=a}^{i-1} y\_m \hat{k}\_{j-i+m} \\ &+ \sum\_{i=b}^N P\_i^+ \sum\_{m=a}^b y\_m \hat{k}\_{j-i+m} \right], \ 0 \leq j \leq N-1. \end{split} \tag{12}$$

*Finally, PN*,1 *can be found from PN*,1 = 1 − *a*−1 ∑ *j*=0 *Pj*,0 − *N*−1 ∑ *j*=0 *Pj*,1.

**Proof.** The fraction of the time the batch server remains idle between two successive departure epochs is the probability of getting the server idle at a random epoch (*Pidle*). Let *E*(*I*) be the mean idle period. Using the definition of *E*(*S*) and *E*(*I*), we have

 $P\_{\text{idle}} = \frac{E(I)}{E(I) + E(S)},$
$$P\_{j,0} = P\_{\text{idle}} \times P(\text{fraction of idle period}) = \frac{E(I)}{E(I) + E(S)} \times \frac{\frac{1}{\lambda} \sum\_{i=0}^{j} P\_i^+}{E(I)}, \ 0 \le j \le a - 1,$$

where *E*(*I*) = 1*λ a*−1 ∑ *i*=0 *P*+*i* (*a* − *i*). We employ system state conditioning and discreet renewal theory to find *Pj*,1. The processor is active with probability (1 − *a*−1 ∑ *i*=0*Pi*,0); thus,

$$P\_{j,1} = P(N\_q = j \text{,processor active}) = (1 - \sum\_{i=0}^{a-1} P\_{i,0})P(N\_q = j \mid \text{processor active}) \tag{13}$$

Assuming that *kej* is the number of jobs that come following an embedded Markov point until the elapsed service time, we have

$$P(N\_q = j \mid \text{processor active}) = \sum\_{i=0}^{a-1} P\_i^+ \sum\_{m=a}^b y\_m \hat{k}\_j + \sum\_{i=a}^{b-1} P\_i^+ \sum\_{m=i}^b y\_m \hat{k}\_j$$

$$+ \sum\_{i=a+1}^{b-1} P\_i^+ \sum\_{m=a}^{i-1} y\_m \hat{k}\_{j-i+m} + \sum\_{i=b}^N P\_i^+ \sum\_{m=a}^b y\_m \hat{k}\_{j-i+m},\tag{14}$$

$$0 \le j \le N - 1. \tag{14}$$

Putting together (13) and (14), we obtain (12). We can obtain *PN*,1 using normalization condition. Thus, we obtain random epoch probabilities {{*Pj*,0}*a*−1 0 , {*Pj*,1}*N*0 } in connection with post departure epoch probabilities {*P*+*j* }*N*0 .

Though the relations between random- and post-departure epoch probabilities are available in [13] using transition rates, here, we develop an alternative method to obtain the queue-length distributions at random epochs.

Because of BASTA (Bernoulli arrivals see time averages) property, see [5], the queuelength distribution exactly before arrival of job will be equal to that of *Pj*,0 and *Pj*,1. Further,

since the outside observation epoch falls in an interval between arrival and departure epochs, the outside observer's distribution is the same as the random epoch distribution.

#### **4. Performance Measures**

This section deals with several measures of performance. The average number of jobs in the queue (*Lq*) is given by

$$L\_q = \sum\_{j=0}^{a-1} j \, \, P\_{\bar{j},0} + \sum\_{j=0}^{N} j \, \, P\_{\bar{j},1}$$

The probability of the processor being busy (PB) at some random moment is specified by 1 − ∑*a*−1 *j*=0 *Pj*,0. Due to the BASTA property, the loss or blocking probability (*PBL*) is given by *PBL* = *PN*,1. Since the effective arrival rate *λe* = *λ*(1 − *PN*,1), we can obtain the average wait time in the queue (*Wq*) by employing Little's law as *Wq* = *Lq λe* . The reported result of *Wq* in [13] is incorrect. They have applied the effective arrival rate as *λ* instead of *λe*.

**Remark 8.** *If ρ* < 1 *and N* → ∞*, then λe* = *λ and Lq* = *a*−1 ∑ *j*=0 *j Pj*,0 + ∞ ∑ *j*=0 *j Pj*,1*. Using Little's law, the average waiting time in the queue (Wq) can be computed as Wq* = *Lqλ .*

#### **5. Numerical Results**

To exemplify the analytic results found in this article, we illustrate several numerical outcomes in tables and figures. We also give several performance measures, for instance, the average queue length (*Lq*), the average waiting time in the queue (*Wq*), and the probability of loss (*Ploss*). The computations were performed in double precision and executed in a 64-bit windows 10 professional OS possessing Intel(R) corei5-6200U processor @2.30 GHz and 8 GB DDR3 RAM utilizing MAPLE 18 software. The numerical results were exact, but we reported outcomes rounding to six decimal places. Because reported outcomes are rounded, the sum of probabilities may not add up to one in some cases.

**Example 1.** *Geo*/*NBa*,*Y*2 /1/10*queue. Consider the distribution of service time as being in two stage negative binomial distribution (NB) with PGF S*(*z*) = *μz* 1−*μ*¯*z*2*. Here, we consider the same parameters as in Table 1 of the paper [13] to compare the results. The parameters are E*(*S*) = 5, *y*2 = 0.2, *y*3 = 0.1*, y*4 = 0.7*, and E*(*Y*) = 3.5*. The arrival rates are* 0.14, 0.35*, and* 0.56 *with corresponding traffic intensities ρ* = 0.2, 0.5*, and* 0.8*, respectively. To show the evaluation process, let us assume λ* = 0.14*. The denominator of Equation* (6) *has six roots, two of which are outside, and four are in and on the unit circle. From Equation* (8)*,*

$$P\_N^+(z) = T \left( 0.068061 + \frac{7.817631}{z - 6.392624} - \frac{6.204397}{z - 5.024572} \right).$$

*where T* = 6.250001*. Similarly, the denominator of P*+*N* (*z*) *has two roots outside the unit circle when arrival rates are* 0.35 *and* 0.56*, and we have from Equation* (8)*,*

$$P\_N^+(z) = T\left(c\_0 + \frac{M\_{b+1}}{z - \gamma\_{b+1}} + \frac{M\_{b+2}}{z - \gamma\_{b+2}}\right),$$

*where γb*+1 = 2.106094*, γb*+2 = 3.454495*, T* = 6.256038*, c*0 = 0.053212, *Mb*+1 = −0.619857*, Mb*+2 = 1.113392 *and γb*+1 = 1.304299*, γb*+2 = 2.688401*, T* = 6.867372*, c*0 = 0.025779*, Mb*+1 = −0.081358*, Mb*+2 = 0.224798*, respectively.*

Now, we can find post-departure epoch probabilities from Equation (9). The results are presented in Table 1. We note that the results of queue-length distribution at post-departure

epoch match the results given by Yi et al. [13], but the random epoch does not. We have also computed queue-length distributions at random epochs using their method for checking purposes. They match perfectly with our results. However, the results presented in the paper by [13] are different. Thus, various performance measures are also not the same.

**Table 1.** Queue-length distributions at various epochs for the *Geo*/*NB*2,*Y*2 /1/10 queue.

**Remark 9.** *It may be noted that P*+*N* (*z*) *is a polynomial in both cases, as can be seen in Table 1. The same applies to other cases as well.*

**Example 2.** *Geo*/*DPHa*,*Y*/1/*Nqueue. The service-time distribution is assumed to be discrete phase-type (DPH) having sk* = *αTk−***1***T***0***, k* = 1, 2, ...*, T***0** = *e* − *Te, where e is the appropriate column vector with all elements equal in size. This gives the PGF of service-time distribution as S*(*z*) = *zα***(***I − zT***)***−***1***T***0***,* |*z*| ≤ 1*. Table 2 shows the queue length distributions at different times employing the DPH service time distribution. For the first example of Table 2, we suppose* 0.200.05

*α* = \$0.40 0.50 0.10%*,* **T** = ⎡⎣0.10 0.30 0.15 0.10 0.20 0.50 0.10⎤⎦*,*

*with E*(*S*) = 2.005267 *and the other parameters are λ* = 0.7, *N* = 15, *y*3 = 0.5, *y*5 = 0.3, *y*8 = 0.2 *with E*(*Y*) = 4.6*. Here the denominator of P*+*N* (*z*) *has eleven distinct roots, out of which three roots are outside the unit circle, and they are γb*+1 = 2.254374*, γb*+2 = −9.788249*, and γb*+3 = −222.447161*. From Equation* (8)*,*

$$P\_N^+(z) = T \left( -3.63007 + \frac{1.301912}{z - 2.254374} - \frac{0.702111}{z + 9.788249} + \frac{930.333318}{z + 222.447161} \right), \tag{15}$$

*where T* = −1.755933*.*

*In the second example of Table 2, we assume α* = \$0.60 0.40%*, and T* = 0.5 1/3 1/3 1/3*with E*(*S*) = 4.2 *and the other parameters are λ* = 0.84, *N* = 50, *y*3 = 0.7, *y*4 = 0.2, *y*5 = 0.1 *with E*(*Y*) = 3.4*. As ρ* = 1.0376*, we have one root in the range* (0, 1) *from the remaining four roots, and the other three are outside the circle of unity. From Equation* (8)*,*

$$P\_N^+(z) = T \left( 0.000674 + \frac{5.2 \times 10^{-21}}{z - 1.375407} + \frac{0.001238}{z - 0.974373} - \frac{0.006324}{z - 16.101113} - \frac{4.900051 \times 10^{-19}}{z - 16.100784} \right), \tag{16}$$

*where T* = −7.569441*.*

**Table 2.** Queue-length distributions at various epochs for the *Geo*/*DPH*3,*Y*/1/15 and *Geo*/*DPH*3,*Y*/1/50 queues.

**Remark 10.** *It may be noted that Equations* (15) *and* (16) *are polynomials since the coefficients of zb are zero. This may be seen in Table 1. This also applies to all the examples that follow.*

**Example 3.** *As before, in this example, we consider two cases: (i) Geo*/*MGeo*2,*Y* 2 /1/10 *queue. Here, the service time is a mixture of two geometric distributions with PGF S*(*z*) = *ς*1 *μ*1*z* 1−(1−*μ*1)*z* + *ς*2 *μ*2*z* 1−(1−*μ*2)*z , where ς*1 + *ς*2 = 1*. The parameters are taken as E*(*S*) = 2.83333, *N* = 10, *ς*1 = 0.6, *μ*1 = 0.4, *μ*2 = 0.3, *y*2 = 0.2, *y*3 = 0.1, *y*4 = 0.7, *E*(*Y*) = 3.5*, λ* = 0.14*. The denominator of P*+ *N* (*z*) *has six distinct roots, out of which two roots are outside the unit circle, and they are γb*+1 = 4.031316 *and γb*+2 = 5.727647*. From Equation* (8)*,*

$$P\_N^+(z) = T \left( -0.561775 - \frac{1.819848}{z - 4.031316} - \frac{4.545280}{z - 5.727647} \right) / 1$$

*where T* = 1.0000001*.*

*(ii) Geo*/*D*2,*Y*/1/50 *queue. Here, we consider service-time distribution as deterministic with PGF S*(*z*) = *zk for some constant k* = 4,*sk* = 1*. The parameters are taken as E*(*S*) = 4, *N* = 50, *y*2 = 0.5, *y*3 = 0.2, *y*4 = 0.1, *y*5 = 0.1, *y*6 = 0.1, *E*(*Y*) = 3.1*, and λ* = 0.4*. In this case, the denominator of P*+ *N* (*z*) *has eight distinct roots, out of which two roots are outside the unit circle, and they are γb*+1 = 5.019701 *and γb*+2 = −11.795711*. From Equation* (8)*,*

$$P\_N^+(z) = T \left( -1.316734z^2 + 1.021799z - 102.664691 + \frac{879.864750}{z + 11.795711} - \frac{141.482007}{z - 5.019701} \right),$$

*where T* = 1*. Table 3 presents queue-length distributions at various epochs when the service-time distributions are a mixture of geometric and deterministic.*

**Table 3.** Queue-length distributions at various epochs for the *Geo*/*MGeo*2,*Y* 2 /1/10 and *Geo*/*D*2,*Y*/1/50 queues.

**Example 4.** *Here, we consider two cases. Table 4 presents the results of Geo*/*Ga*,*Y*/1/∞ *which can be found from the Geo*/*Ga*,*Y*/1/*N system by assuming ρ* < 1 *and buffer capacity N appropriately large. We can easily compute the queue-length distributions of infinite queue capacity from finite queue capacity by assuming ρ* < 1 *and N sufficiently large.*

*(i) Geo*/*NBa*,*Y*/1/∞ *queue. Here, we assume negative binomial (NB) service time distribution and the parameters are taken as λ* = 0.703, *E*(*S*) = 5, *y*2 = 0.5, *y*4 = 0.2, *y*5 = 0.1, *y*6 = 0.1, *y*8 = 0.1 *with E*(*Y*) = 3.7*. From Equation* (8)*, we have*

$$P\_N^+(z) = T \left( 0.005129 + \frac{0.03006}{z - 2.520637} - \frac{0.00745}{z - 1.042657} \right) \text{s}$$

*where T* = 6.250024*. (ii) Geo*/*DPHa*,*Y*/1/∞ *queue. For a DPH service time distribution, the settings are chosenas α* = \$0.60 0.40% *and* **T** = 0.5 1/3 1/3 1/3 *.*

*with E*(*S*) = 4.2 *and the other parameters are λ* = 0.58, *y*2 = 0.5, *y*3 = 0.3, *y*4 = 0.2*, and E*(*Y*) = 2.7*. From Equation* (8)*, we have*

$$P\_N^+(z) = T \left( -0.001463 + \frac{0.019982}{z - 22.872839} - \frac{3.590919 \times 10^{-14}}{z - 22.870101} - \frac{0.004169}{z - 1.080550} + \frac{1.09 \times 10^{-20}}{z - 1.543692} \right),$$

*where T* = 20.250*.*

**Table 4.** Queue-length distributions at various epochs when *N* → ∞.

In Figure 2, we compare the processing times to calculate probabilities at post-departure using the proposed technique and the method used by [13] (solving a linear system of equations (SLSE)) against finite buffer capacity. We take the NB service-time distribution in two stages with the input parameters in the same way as in Table 1 for *ρ* = 0.2. We notice that, with the increase of *N*, the time needed by the roots method remains almost static, whereas the method used by [13] takes more time for larger buffer size *N* and increases remarkably as *N* increases. In the roots method, the variation of the time is minimal. This is because of the initial estimation of the Newton iterative method for the calculation of polynomial roots. The application of the SLSE process is low in reliability and time-consuming. The root method provides a faster solution and superior performance in solving a linear system of method equations both in speed and reliability.

**Figure 2.** Time (in seconds) needed to calculate post-departure probabilities.

Figure 3 shows the roots of the characteristic equation for the number in the queue with NB service time distribution having 4 successes is the convolution of 4 geometric distributions. Here we consider the parameters as *λ* = 0.81, *y*4 = 0.4, *y*6 = 0.1, *y*12 = 0.1, *y*25 = 0.2, *y*30 = 0.1, *y*36 = 0.1, and *ρ* = 0.54. There are 40 of roots inside, on, and out of the unit circle for the assumed parameters. Here the characteristic equation is

$$\begin{aligned} z^{36} \big(-0.886 + 0.486z\big)^{4} - 0.0256(0.19 + 0.81z)^{4} \big(0.1 + 0.4z^{32} + 0.1z^{30} \\ + 0.1z^{24} + 0.2z^{11} + 0.1z^{6}\big). \end{aligned} \tag{17}$$

**Figure 3.** The 40 roots of Equation (17) when NB service-time distribution.

## **6. Conclusions**

This article focuses on the *Geo*/*Ga*,*Y*/1/*N* queue length distributions at various points in time. We use the roots of the associated characteristic equation to determine a unified way to compute performance measures for both infinite- and finite-buffer systems. Queue length distributions at a post-departure time are computed using an embedded Markov chain method. We obtain associations between queue length distributions at several time points by applying system state conditioning and discrete renewal theory. Several performance indices have been carried out, such as the blocking probability, the average wait time in the queue, and the average number of jobs in the queue. We illustrate them by using different numerical outcomes. The approach discussed in this paper can be used to cover cases when customers arrive in groups or even when arrivals are correlated (D-MAP-discrete Markovian arrival process).

**Author Contributions:** Conceptualization, M.C. and V.G.; methodology, V.G. and M.C.; Writing— original draft preparation, V.G.; supervision, M.C.; funding acquisition, M.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by FMAS IO 757193 (C143).

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. The Continuous-Time Case**

Here, we consider the relation between the discrete-time *Geo*/*Ga*,*Y*/1/*N* queue and its continuous-time analogue. Let the time axis be divided into periods of uniform length Δ*u* with Δ*u* > 0 sufficiently small. In *Geo*/*Ga*,*Y*/1/*N*, since the inter-arrival times (*u*) follow geometric distribution, the arrivals will follow binomial distribution which, as we have seen earlier, leads to PGF for binomial distribution. In the continuous-time case, geometric tends to exponential, and binomial tends to Poisson distribution, and the PGF tends to

Laplace transform. Let us discuss this analytically. Assume that the inter-arrival times in the case of *M*/*Ga*,*Y*/1/*N* have a rate *α*. Then, *λ* = *α*Δ*u* + *o*(Δ*u*). In the discrete-case, let the service times *S* be in multiples of Δ*u* with probability *P*(*S* = -Δ*u*) = *s*- and ∞ ∑ -=1 *s*- = 1. Further, let *n*Δ*u* = *vn*, where the interval [0, *vn*] is divided into *n* intervals of length Δ*u*. The PGF of an arrival (or no arrival) in the interval (*v*-, *v*-+1) is (1 − *λ* + *λz*). If we denote the probability density function of service times by *h*(·), then

$$\begin{aligned} P(\text{service finishes in } (\iota, \iota + \Delta \iota) \mid \text{service time} > \iota) &= h(\iota) \Delta \iota + o(\Delta \iota) \\ \text{and} \quad s\_\ell &= h(v\_\ell) \Delta \iota + o(\Delta \iota). \end{aligned}$$

When Δ → 0, the PGF *S*(1 − *λ* + *λz*) changes to a Laplace transform. Using the definition of Lebesgue integration and taking the limit as Δ → 0 and *λ* = *α*Δ, we have

$$\lim\_{\Delta \to 0} K(z) = \lim\_{\Delta \to 0} \sum\_{\ell=1}^{\infty} s\_\ell (1 - \lambda + \lambda z)^\ell = \bar{h}(\alpha - \alpha z).$$

The proof of the above is not discussed in detail here since the method applied can be found in [28]. Now, using *λ* = *α*Δ, *K*(*z*) = ¯ *h*(*α* − *αz*) in (5) and taking the limit as Δ → 0, we have

$$P\_N^+(z) = \frac{\bar{h}(a - az) \left[ \sum\_{i=0}^{a-1} P\_i^+ \left( z^b - z^i Y'(z) \right) + \sum\_{i=a}^{b-1} P\_i^+ \left( \sum\_{r=i}^b y\_r z^b - z^i \sum\_{r=i}^b y\_r z^{b-r} \right) \right]}{z^b - \bar{h}(a - az) Y'(z)},$$

the connections for *M*/*Ga*,*Y*/1/*N* system. Taking *a* = 1, we have

$$P\_N^+(z) = \frac{h(\alpha - \alpha z) \left[ \sum\_{i=0}^{b-1} P\_i^+ z^b \left( \sum\_{r=i}^b y\_r - z^i \sum\_{r=i}^b y\_r z^{-r} \right) \right]}{z^b - \tilde{h}(\alpha - \alpha z) Y'(z)},$$

which matcheswith [29].

Note that we cannot obtain results of discrete-time analogue from [29], that is, the converse is not true.


**3. Results**

### *3.1. The Orthologs of WC-1 and WC-2 in F. asiaticum*

Since *F. asiaticum* belongs to a sub-lineage of the *Fusarium graminearum* species complex, the orthologous genes of *wc1* and *wc2* in *F. asiaticum*, namely *Fawc1* and *Fawc2*, were identified as follows: BLASTP search of the *F. gaminearum* genome in Ensembl Fungi database, using the amino acid sequences of *Neurospora crassa* WC-1 (NCU02356) and WC-2 (NCU00902) proteins [34] as queries, resulted in the corresponding orthologs namely FgWC1 (FGSG\_07941) and FgWC2 (FGSG\_00710), respectively. The corresponding genes and their flanking sequences were used as references to design specific primers to amplify the open reading frame (ORF) sequences of their orthologs, *Fawc1* and *Fawc2*, in *F. asiaticum* strain EXAP-08 via the *Pfu* DNA polymerase. Sequencing of the cloned PCR products indicated that *Fawc1* (KX905081.1) and *Fawc2* (MT019868) of *F. asiaticum* showed high similarity to *Fgwc1* (99.05%) and *Fgwc2* (99.39%) of *F. graminearum*. The deduced amino acid sequences of FaWC1 and FaWC2 were analyzed via the SMART online tool, and the results indicated that FaWC1 possessed a Zn-finger DNA-binding domain as well as three PAS domains, of which the N-terminal most PAS domain should be the special subclass called the LOV domain (for light, oxygen, and voltage) being responsible for binding the chromophore molecule of flavin adenine dinucleotide (FAD), while the FaWC2 just essentially contained a single PAS domain and a Zn-finger DNA binding domain. In general, FaWC1 and FaWC2 show high similarity with their orthologs in *F. graminearum* and *N. crassa* (Figure 1A). Gene expression analysis showed that both *Fawc1* and *Fawc2* were induced to peak levels by 15 min light exposure, while the long period (12 h) of illumination caused less induction of the transcription levels of these two genes (Figure 1B).

**Figure 1.** Identification of two photoreceptor genes, *Fawc1* and *Fawc2*, in *F. asiaticum*. (**A**). Schematic demonstration of the domains of WC1 and WC2 orthologs from *N. crassa*, *F. graminearum*, and *F. asiaticum*. Accessions of the amino acid sequences are as follows: WC1 (NCU02356); WC2 (NCU00902); FgWC1 (FGSG\_07941); FgWC2 (FGSG\_00710); FaWC1 (KX905081.1); FaWC2 (MT019868). Domains of these photoreceptor proteins were analyzed via the SMART online tool (http://smart.embl-heidelberg.de/). PAS: Per-period circadian protein; Arnt: Ah receptor nuclear translocator protein; Sim: Single-minded protein, NLS: Nuclear Location Singal, Zn: Zinc finger binding to DNA consensus sequence. (**B**). Transcript levels of *Fawc1* and *Fawc2* are regulated by light. The horizontal axis indicates light treatment time. The bars present mean values ± SD of three replicate samples.

### *3.2. Generation and Characterization of the* Δ*Fawc1 and* Δ*Fawc2 Mutants*

To reveal the functions of *Fawc1* and *Fawc2*, homologous recombination cassettes used for gene knockout purposes were created as shown in Figure 2. After protoplast transformation, the transformants with hygromycin resistance successfully grew up on the selection medium. To characterize the knockout mutants of Δ*Fawc1* and Δ*Fawc2* in which the hygromycin resistance cassette (*hph*) had correctly replaced each target gene, PCR assays with the specific primer pairs were performed with the genomic DNA, and the correct transformants for each gene were selected for further analysis. To verify that the *Fawc1* or *Fawc2* was completely knocked out, reverse transcription (RT)-PCR was applied, revealing that the transcripts of *Fawc1* and *Fawc2* were present in the EXAP-08 wild-type strain but absent in the mutants of Δ*Fawc1* and Δ*Fawc2*, respectively. To confirm the functions of these two genes, the wild-type *Fawc1* and *Fawc2* connected with their native promoters, and terminators were transformed into Δ*Fawc1* and Δ*Fawc2* to obtain the complementation strains. As FaWC1 possesses both signal input and output domains, the LOV and zinc finger (ZnF) domains, respectively, the truncated versions of FaWC1, lacking either LOV or ZnF domains (Figure 2), were expressed in the Δ*Fawc1*

mutant to explore the functions of these domains in light signaling and other life aspects of *F. asiaticum*. All the fungal strains used in this study and their genotypes are listed in Table 1.

**Figure 2.** Generation of the transgenic mutants of *F. asiaticum.* Schematic diagrams of homologous recombination occurred between the replacement vector carrying the hygromycin resistance marker (*hph*) and the target gene (*Fawc1*) of *F. asiaticum* strain EXAP-08, resulting in the knockout mutant (Δ*Fawc1*). Meanwhile, the wild-type *Fawc1* or its truncated versions, *Fawc1*ΔLOV and *Fawc1*ΔZn, were amplified from genomic DNA of EXAP-08 and cloned into the flu6 plasmid, and then transformed into the Δ*Fawc1* mutant, resulting in the complementation strain Δ*Fawc1*-*C*, and Δ*Fawc1-C*ΔLOV and Δ*Fawc1-C*ΔZnF mutant strains. The knockout mutant Δ*Fawc2* and its complementation strain Δ*Fawc1*-*C* were generated via a similar strategy.

### *3.3. The Marker Responses to Light Signal Are Mediated by WCC and Dependent on LOV but not ZnF Domain of FaWC1 in F. asiaticum*

As fungi commonly sense light to indicate the presence of deleterious ultraviolet (UV) radiation, the light-induced survivability through UV damage is recognized as a marker response to light signal in fungi [35,36]. In the present study, the survival of serially diluted spores of all the tested strains was similarly poor if the fungal cultures were kept in the dark after UV-C treatment. However, when white light illumination was applied, the wild type (WT) could survive much better from UV-C irradiation than those cultures in the dark, but no recovery of survival spores was observed with Δ*Fawc1* and Δ*Fawc2*. The complementation strains, Δ*Fawc1-C* and Δ*Fawc2-C*, demonstrated similar photoreactivation levels as the WT, indicating that the defect in the light-induced UV-C resistance in the Δ*Fawc1* and Δ*Fawc2* mutants is indeed caused by the loss of these two genes (Figure 3A). Interestingly, the Δ*Fawc1-C*ΔLOV showed similar UV-C susceptibility as the Δ*Fawc1* and Δ*Fawc2* mutants, while in contrast, the Δ*Fawc1-C*ΔZnF restored UV-C tolerance to the WT level. Gene expression analysis showed that the transcript level of the photolyase gene Faphr1 was significantly induced by light in WT, Δ*Fawc1-C*, Δ*Fawc2-C*, and Δ*Fawc1-C*ΔZnF strains; however, no remarkable change of *Faphr1* expression as influenced with light has been observed in the Δ*Fawc1*, Δ*Fawc2*, and Δ*Fawc1-C*ΔLOV strains (Figure 3B).

**Figure 3.** Effect of light on UV-C resistance. (**A**). Serial dilutions of all strains were point-inoculated onto complete agar medium (CM). After the UV irradiation of indicated dosages, the plates were incubated for one day in light (left) or darkness (right). (**B**). The relative expression level of the deduced photolyase gene *Faphr1* in wild-type and mutant strains as influenced by light. DD, samples cultured for 48 h in darkness; LL, samples experienced 47 h culture in darkness followed by one hour of light illumination. The bars present mean values ± SD of three replicate samples.

Another marker response to the light signal in fungi is pigment production [18,37]. The WT strain could produce significantly more orange-colored carotenoid pigment in constant light compared to dark condition after growth in liquid complete medium (CM) for three days (Figure 4A). In contrast, there was no observable orange pigment accumulated by Δ*Fawc1* and Δ*Fawc2* in light and dark conditions (Figure 4A). Similarly, the Δ*Fawc1-C*ΔLOV mutant, which lacks the LOV domain of FaWC1, showed similar pigmentation phenotypes as the Δ*Fawc1* and Δ*Fawc2* mutants. However, the Δ*Fawc1-C*ΔZnF with truncating the ZnF domain of FaWC1 demonstrated enhanced carotenogenesis in response to light, which was similar to the WT strain. Quantitative measurement assay also showed that in darkness, all strains produced basic carotenoid levels, and light treatment caused a significant increment of carotenoid accumulation in WT, Δ*Fawc1-C*, Δ*Fawc2-C*, and Δ*Fawc1-C*ΔZnF strains, but light failed to alter the pigmentation behavior in Δ*Fawc1*, Δ*Fawc2-C*, and Δ*Fawc1-C*ΔLOV strains (Figure 4B). Gene expression analysis showed that the transcript levels of the carotenoid biosynthetic genes *CarRA* and *CarB* [37,38] were up-regulated in light versus dark condition in the WT, ΔFawc1-C, ΔFawc2-C, and Δ*Fawc1-C*ΔZnF strains. Contrarily, the expression of *CarRA* and *CarB* could not be induced by light in the Δ*Fawc1*, Δ*Fawc2*, and Δ*Fawc1-C*ΔLOV strains (Figure 4C,D).

**Figure 4.** Effect of light on carotenogenesis. (**A**). Carotenoid pigment accumulation of the tested strains cultured in liquid CM for four days under constant light (LL) or darkness (DD). (**B**). Measurement of carotenoid contents in the mycelium of each strain harvested from the liquid shaking culture in (**A**). (**C**) and (**D**). Relative expression levels of deduced carotenoid biosynthesis genes *CarRA* and *CarB* in wild-type and mutant strains under light (LL) or darkness (DD). The bars in **B**, **C**, and **D** present mean values ± SD of three replicate samples.

Collectively, the above data suggest that both FaWC1 and FaWC2 are responsible for light signaling to induce the marker responses, including carotenoid accumulation and UV damage tolerance. Moreover, the LOV and ZnF domains of FaWC1 are required and dispensable, respectively, for mediating the light responses in *F. asiaticum*.

### *3.4. Perithecia Maturation and Ascospore Development of F. asiaticum Are Regulated by WCC Photoreceptor*

Sexual reproduction is usually vital for the dissemination of fungal pathogens in their lifecycles, and the near-UV light is known to induce the perithecia maturation and ascospore formation in FGSC [39]. However, two independent studies reported inconsistent e ffects of the WCC photoreceptor on the sexual development of *F. graminearum*, which was probably due to the di fference of the wild-type background strains used in each study [30,40]. The present work with *F. asiaticum* showed that the WT strain was able to form mature perithecia, in which the sexual ascospores could be found apparently. In contrast, the Δ*Fawc1* and Δ*Fawc2* mutants could produce a comparable amount of perithecia as WT, but these mutants' perithecia failed to develop into the black-pigmented mature stage, being deficient in ascospore formation (Figure 5). Re-introducing the wild-type *Fawc1* and *Fawc2* into the corresponding gene deletion mutants had fully recovered their perithecia maturation and ascospore formation abilities, suggesting that these WCC photo receptor components, FaWC1 and FaWC2, are indeed required for sexual reproduction development in this fungus. Additionally, the Δ*Fawc1-C*ΔZnF and Δ*Fawc1-C*ΔLOV demonstrated phenotypes in sexual development as WT and Δ*Fawc1*, respectively, indicating that it should be the LOV domain, but not the ZnF domain, that is required for mediating the light signal to regulate the sexual reproduction of *F. asiaticum*.

**Figure 5.** Regulation of *Fawc1* and *Fawc2* on the sexual reproduction development processes of *F. asiaticum*. The perithecium formed by each strain on carrot agar medium was observed via stereomicroscope. Via picking up the perithecia and pressing them on glass slides for microscopic analysis, mature ascospores were observed from the perithecia of EXAP-08, Δ*Fawc1-C*, Δ*Fawc1-C*ΔZnF, and Δ*Fawc2-C* strains, while in contrast, only mycelium biomass could be found in the immature perithecium of Δ*Fawc1*, Δ*Fawc2*, and Δ*Fawc1-C*ΔLOV mutant strains.

### *3.5. FaWC1 and FaWC2 Play Di*ff*erent Roles in Regulating Virulence Expression*

In the infection assay with wheat coleoptiles, inoculation with the WT caused apparent brown rot symptom in the host plant materials (Figure 6). In contrast, the Δ*Fawc1* mutant showed more than 80% reduction in pathogenicity in comparison with WT. Meanwhile, the complementation strain Δ*Fawc1-C* had a recovered pathogenicity level similar to that of the WT strain, suggesting that FaWC1 is involved in regulating the pathogenicity of *F. asiaticum*. However, the mutant with the deletion of *Fawc2* caused equivalent disease severity to the WT. These data suggested that FaWC1 and FaWC2 played independently different roles in regulating the pathogenicity of this fungus.

**Figure 6.** Virulence assay on wheat confirms that *Fawc1* regulate the virulence of *F. asiaticum* in a light-independent manner. (**A**). Wheat coleoptiles were inoculated with 5 μL conidial suspensions and were kept humid inside a plastic box. Fungal strains for test include the wild-type EXAP-08, Δ*Fawc1*, Δ*Fawc1*-*C*, Δ*Fawc1-C*ΔLOV, Δ*Fawc1-C*ΔZnF, Δ*Fawc2*, and Δ*Fawc2*-*C*. Photographs were taken at four days post inoculation. (**B**). Statistical analysis of lesion sizes caused by each fungal genotype. Different letters represent a significant difference at *p* < 0.05. The bars present mean values ± SD (*n* = 20).

In order to assess whether and how light signaling via the WCC pathway could be involved in fungal pathogenicity expression, the mutant strains lacking the LOV and ZnF domains of FaWC1 were further analyzed in the host infection assay. Unexpectedly, the Δ*Fawc1-C*ΔLOV mutant showed similar

pathogenicity as the WT and complementation strains. While in contrast, the pathogenicity of the Δ*Fawc1-C*ΔZnF mutant was similar to the Δ*Fawc1* mutant, being significantly reduced compared to WT (Figure 6). Consequently, it can be concluded that the FaWC1 LOV domain, which is required for sensing light signals, exerts no influence on pathogenicity; on the other hand, the ZnF domain of FaWC1 is involved in regulating pathogenicity, although this domain is dispensable for mediating light signals in *F. asiaticum*.


*4.2. Band Selection Using Optimization Algorithms (PF2 and PF3)*

The *PF2* and *PF3* aim to use optimization algorithms in order to find the most relevant bands able to perform an accurate classification of the brain tumors, using the lower possible number of features. The evaluation of both processing frameworks was performed using the reduced training dataset, and a data partition scheme following a leave-one-patient-out cross-validation to create the SVM model and evaluate the results (see Figure 3b,c). In these processing frameworks, the six test HS images were evaluated with different optimization algorithms to find out which offers the best results. In addition, in case of *GA* and *PSO* algorithms (PF2), two different metrics were employed to evaluate the selected bands: *OAPenalized* (OA\_P) and *FoMPenalized* (FoM\_P). In summary, the band selection techniques evaluated were: *GA* using *OAPenalized* (PF2-GA-OA\_P); *PSO* using *OAPenalized* (PF2-PSO-OA\_P); *GA* using *FoMPenalized* (PF2-GA-FoM\_P); *PSO* using *FoMPenalized* (PF2-PSO-FoM\_P) and *ACO* using 60 bands (PF3-ACO-60). Furthermore, all the results were compared with the reference values obtained with the *PF1* using 128 bands (PF1-128).

Figure 7 shows the boxplot results of the OA and the normal, tumor and hypervascularized tissue sensibilities obtained after the evaluation of the *GA*, *PSO* and *ACO* algorithms. The *ACO* algorithm was evaluated selecting different numbers of bands (20, 40, 60, 80 and 100), but only the results obtained with 60 bands have been reported because they were found to be the most competitive ones. Figures S3 and S4 in the supplementary material present the detailed results obtained with the *ACO* algorithm. Figure 7a shows the OA results of all the techniques, where it should be noted that the median values are around 80%, offering the PF2-GA-OA\_P the best result. However, the results obtained in the tumor sensitivity boxplot (Figure 7b) shows that the technique that uses the *GA* algorithm with the *FoMPenalized* metric (PF2-GA-FoM\_P) provides the best results, achieving a tumor sensitivity median of ~79%. This represents an increment of ~21% with respect to the PF2-GA-OA\_P method. On the other hand, in Figure 7c, it can be seen that the results of the normal tissue sensitivity are similar in both cases (PF2-GA-OA\_P and PF2-GA-FoM\_P), with a median value of 89% and 90%, respectively. The same behavior can be observed in the hypervascularized tissue sensitivity results (Figure 7d). As can be observed in Figure 8, the MCC metric, which takes into account the unbalance of the test labeled dataset, shows the same behavior as the other metrics. Thus, the PF2-GA-FoM\_P is the method that provides on average the best results. This is especially highlighted in the tumor class results.

Figure 9 illustrates the qualitative results represented in the classification maps obtained for each method. These maps allow visualization of the identification of the different structures for each class found in the complete HS cube, i.e., it is possible to visually evaluate the results obtained in the non-labeled pixels of the HS test cubes. In addition, this figure indicates the number of bands selected with each method for each HS test cube. Figure 9a shows the synthetic RGB images for each HS test cube, indicating the location of the tumor area surrounded by a yellow line, while Figure 9b shows the classification results obtained with the reference method (PF1-128).

**Figure 7.** Boxplot results of the leave-one-patient-out cross-validation obtained for each processing framework. (**a**) Overall accuracy. (**b**) Tumor tissue sensitivity. (**c**) Normal tissue sensitivity. (**d**) Hypervascularized tissue sensitivity.

**Figure 8.** Boxplot results of the Matthews correlation coefficient (MCC) metric using the leave-one-patient-out cross-validation obtained for each processing framework. (**a**) Tumor tissue. (**b**) Normal tissue. (**c**) Hypervascularized tissue.

**Figure 9.** Classifications maps of the test database. (**a**) Synthetic RGB images with a yellow line identifying the tumor area. (**b**) Reference results using 128 bands. (**c**) Results of the *GA* algorithm using *OAPenalized*. (**d**) Results of the *PSO* algorithm using *OAPenalized*. (**e**) Results of the *GA* algorithm using *FoMPenalized*. (**f**) Results of the *PSO* algorithm using *FoMPenalized*. (**g**) Results of the *ACO* algorithm using the 60 bands per image.

Considering these images as reference, it is observed that the results in all cases are very similar. Nonetheless, when analyzing the images one-by-one, in the case of the first image, *P008-01*, it is observed that using either PF2-PSO-OA\_P (Figure 9d) or PF3-ACO-60 (Figure 9g) the results are more accurate with fewer false positives. The best case that visualizes the *P008-02* image is the PF2-GA-FoM\_P (Figure 9e), since it is observed a higher number of tumor pixels in the area of the tumor. As for the *P012-01*, all the techniques visualize the tumor area correctly, but the PF1-128 (Figure 9b) is the one that shows fewer false positives pixels. In the case of *P012-02*, the best technique is the PF2-GA-FoM\_P (Figure 9e) due to it shows less false positive pixels. With respect to the *P015-01* image, it is observed that using PF2-GA-OA\_P (Figure 9d) and PF2-GA-FoM\_P (Figure 9e) the tumor area is clearly identified, although they have a small group of false positives in the upper left image due to some extravasated blood out of the parenchymal area. Finally, image *P020-01* offers practically the same result in all cases without a successful identification of the tumor area, even in the reference results. Regarding the number of bands selected to perform the classification, the PF2-GA-OA\_P and PF2-GA-FoM\_P are the methods that achieved the less number of bands for each HS test image, being lower than 18 bands in all the cases.

After conducting a thorough analysis of the results obtained, it was observed that the best technique, which provided the best balance between qualitative and quantitative results, is the PF2-GA-FoM\_P. Quantitatively, the GA-FoM\_P provided the best average OA value of 78% (improving ~4% with respect to the reference results with 128 bands) and the best median tumor sensitivity value of 79%, which represents an increment of ~21% with respect to the best solution provided with the other optimization approaches. Moreover, the GA-FoM\_P increases the tumor sensitivity value in 30% with respect to the reference results.