Analysis of the M/G/1 queue in multi-phase random environment with disasters

Abstract

This paper studies an M/G/1 queue in a multi-phase random environment. When in operative phase i, i = 1, 2, . . . , n, the system is subject to disastrous interruptions, causing all present customers (waiting and served) to leave the system. At an exponential failure instant, the server abandons the service and the system goes directly to repair phase. After an exponential repair time, the system moves to operative phase i with probability q i , i = 1, 2, . . . , n. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive results of the cycle analysis, the sojourn time distribution and the length of the server's working time in a service cycle. In addition, some special cases and numerical examples are presented.

On the GI/M/1 Queue with Vacations and Multiple Service Phases

“…. , , then {( ℓ , ℓ ), ℓ ≥ 1} is positive recurrent; thus we have (21) and (22). For ≥ 2, using equation (1.5.4) of Neuts [24], we obtain…”

“…In their 2 Mathematical Problems in Engineering model, the system moves from repair phase to some service environment with a certain probability after being repaired. Jiang et al [22,23] extended Paz and Yechiali's work to an / /1 and / /1 queue. In [15], Li and Liu studied a discrete-time / /1 queue with vacations in random environment in which the system jumps from vacation phase to a service phase with a certain probability when vacation phase ends.…”

Service Rate Optimization of Finite Population Queueing Model with State Dependent Arrival and Service Rates

“…Haviv and Oz [9] reviewed some existing observable queueing mechanisms where money transfers was taken into account concluding that the best ones are those in which customers have to make up their mind to join the queue without inspecting the queue length. Jiang et al [10] dealt with a disaster queue in a multi-phase random environment in which the system stops working suddenly and resumes after exponential repair time. Kumar et al [11] derived an optimization model of an feedback queue with retention of reneged customers.…”

Analysis of an M/G/1 Stochastic Clearing Queue in a 3-Phase Environment

“…Lemma 2 (see Lemma 2 in [12]) Let X, Y, Z be the random variables, and Z = min (X, Y ), where X follows a general (arbitrary) distribution with mean 1 µ and distribution function X(ν) and Laplace Stieltjes transform X * (s), Y follows an exponential distribution with parameter η, then we have…”

“…In this paper, we give the stationary queue size distribution at an arbitrary epoch. Following the idea presented by Jiang et al [12] , which provided an approach to analyze the sojourn time distribution and the length of the server's working time in a cycle, we also derive these performance measures in our paper. The rest of this paper is organized as follows: Section 2 is the model description.…”