the geometric PushTASEP one can pass to the continuous time by sending $\mu\to0$ and rescaling the discrete time by $\mu^{-1}$. For $q>0$, this produces the continuous time $q$-PushTASEP introduced in [@BorodinPetrov2013NN] and considered in [@CorwinPetrov2015]. For $q=0$ the process reduces to the usual continuous time PushTASEP [@Spitzer1970], [@derrida1991dynamics] (also referred to as the “long-range TASEP”, and viewed as a simplified model of Toom’s anchored interface dynamics).

In the geometric $q$-PushTASEP from [@MatveevPetrov2014], as well as in its degenerations, one can assign update probability with the particle-dependent parameter $\mu_i$ to each $x_i$, and the resulting system remains exactly solvable via $q$-Whittaker or Schur symmetric functions (for the $q$-PushTASEP and PushTASEPs, respectively). Moreover, in the $q$-Hahn TASEP one can also take particle-dependent parameters $(\mu_i,\nu_i)$, under the condition that $\nu_i/\mu_i$ does not depend on the particle. The $q$-moments of the resulting system can be expressed as contour integrals coming from the inhomogeneous stochastic higher spin six vertex model [@BorodinPetrov2016inhom Section 10.3]. It is likely that the $q$-Hahn PushTASEP duality and moment formulas can be extended to include particle-dependent parameters, but here we do not pursue this direction.[^4]

### A new $q=0$ particle system {#sub:gB_push}

Let us briefly describe a limit as $q\searrow0$ and 