Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

Abstract

In this article we analyze the M/G/1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M/M/1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

“…The load of priority customers is ρ 1 = λ 1 β 1 1 . Similarly, the LST of B 2 (x) is denoted as B 2 (s), the first moment of B 2 (x) is β 2 1 , and the load of ordinary customers is given by ρ 2 = λ 2 β 2 1 . Note that inter-arrival times of primary customers, intervals between repeated trials, and service times are assumed to be mutually independent.…”

“…Theorem 5.1. The transition operator of the embedded Markov chain {X d } is monotone with respect to the order ≤ so , that is, for any two distributions ϕ (1) and ϕ (2) , the inequality ϕ (1) ≤ so ϕ (2) implies that Θϕ (1) ≤ so Θϕ (2) , where ≤ so =≤ st or ≤ v .…”

“…In this section, we incorporate the transition operators Θ (1) and Θ (2) into models Σ (1) and Σ (2) , respectively. The following theorem provides comparability conditions for two embedded Markov chains.…”

“…then Θ (1) ≤ so Θ (2) , that is, for any distribution ϕ, one has Θ (1) ϕ ≤ so Θ (2) ϕ, where ≤ so is either ≤ st or ≤ v .…”

“…Additionally, it has been employed in studies of partially ordered spaces [35] and applied probability [47]. Several previous works have also applied the stochastic comparison method to queueing models, including [1,2,[7][8][9][10][11][12][13][14][15][16]34].…”