q-SERIES IN MARKOV CHAINS WITH BINOMIAL TRANSITIONS

Abstract

We consider a single-server Markovian queue with synchronized services and setup times. The customers arrive according to a Poisson process and are served simultaneously. The service times are independent and exponentially distributed. At a service completion epoch, every customer remains satisfied with probability p (independently of the others) and departs from the system; otherwise, he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty.Some of the transition rates of the underlying two-dimensional Markov chain involve binomial coefficients dependent on the number of customers. Indeed, at each service completion epoch, the number of customers n is reduced according to a binomial (n, p) distribution. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period, and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of such systems.

Synchronized reneging in queueing systems with vacations

“…The new feature of these models with synchronization is the existence of binomial type jumps at the abandonment epochs. Similar models with binomial type transitions have been recently studied by Economou (2004), Economou and Kapodistria (2006), Artalejo et al (2007) and Economou and Fakinos (2008).…”

Linear Birth/Immigration-Death Process With Binomial Catastrophes

“…Several other papers deal with the equilibrium analysis of birth models subject to binomial catastrophes (see, e.g. [2,5,20,22,23,24,25,38,40] and the references therein).…”

“…In [22], for a different birth model still subject to binomial catastrophes, the authors discuss possible extensions of the existing methods in order to calculate the transient distribution of the system at hand. Several other papers deal with the equilibrium analysis of birth models subject to binomial catastrophes (see, e.g., [2,5,20,[22][23][24][25]38,40] and references therein).…”

Asymptotics for the late arrivals problem

“…To the best of our knowledge, the functional equation 14below has never previously appeared in the literature. Yet it is possible to mark some analogies with the functional equation studied by Economou and Kapodistria (2009), Economou et al (2010),or Kapodistria (2011), the most important being that in both equations the right hand side exhibits the generating function computed in a convex combination in the parameter q. Adan et al (2009), Altman and Yechiali (2006), Neuts (1994), and Yechiali (2007) give other examples of chains with binomial transitions.…”