STABILITY OF MAP/PH/c/K QUEUE WITH CUSTOMER RETRIALS AND SERVER VACATIONS

Abstract

Abstract. We consider the M AP/P H/c/K queue in which blocked customers retry to get service and servers may take vacations. The time interval between retrials and vacation times are of phase type (PH) distributions. Using the method of mean drift, a sufficient condition of ergodicity is provided. A condition for the system to be unstable is also given by the stochastic comparison method.

Dimension Reduction for Approximation of Advanced Retrial Queues : Tutorial and Review

“…Breuer et al [9] indicate that the proof of the sufficient condition for K = c is not correct. However, Shin [49] showed that the result of [25] is correct . Approximation methods for the system with non-exponential retrial times are developed for M/G/1 retrial queue with general retrial time [67], for M/G/1 retrial queue with mixture of Erlang retrial time [35] and for M/P H/1 retrial queue with P H-retrial time [15].…”

“…) is a continuous time Markov chain on the state space S = {(n n n, k, j) ∈ Z g+2 + : n n n ≥ 0, 0 ≤ k ≤ c, 0 ≤ j ≤ w} and Ψ Ψ Ψ is positive recurrent if ρ = λ cµ < 1 [49]. Let (X X X, Y, J) be the stationary version of Ψ Ψ Ψ.…”

“…The call center with outgoing calls introduced in [43] can be considered as the queues with retrials and vacations. An algorithmic solution for the M AP/M/c/K queue with PH-vacation time and exponential retrial time is developed in [48] and the stability condition for the M AP/P H/c/K queue with P H vacation time and P H retrial time is given in [49].…”

“…There are a few qualitative approach for the multi-server retrial queues in the third class, e.g. a stability condition for BM AP/P H/s/s + K retrial queue with P H retrial time [25], a stability condition for M AP/P H/s/K retrial queue with P H retrial time and server vacation [49] and GI/G/c retrial queue with exponential retrial time [39] and the monotonicity for M/G/1 retrial queue with general retrial time [36] and for A X /B/c/K retrial queue with exponential retrial times [44]. The retrial queues with structurally complex structure have been known to be difficult for mathematical analysis because the joint queue length process is a random walk on the multidimensional integer lattice even it is modelled by a Markov chain.…”

Multiserver retrial queue with setup time and its application to data centers

“…The stability condition of our multiserver retrial queue is a new finding in the sense that it is different from that of the corresponding queue without retrials. In this line, we recently became aware of Shin [24] in which a stability condition is derived for MAP/PH/c/K retrial queues with vacation. It should be noted that vacation model is essentially different from setup model because in the former a vacation is independent of waiting customers while in the latter OFF servers are activated by waiting jobs.…”