This paper considers a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.

Quasi-birth-and-death processes, lattice path counting, and hypergeometric functions

Bi-objective optimization of a queueing model with two-phase heterogeneous service

ABSTRACT A retrial queue with constant retrial policy, unreliable servers and the failed servers may not be detected is investigated in this paper. Upon arrival, a customer either is attended to one of the available servers or enters a retrial orbit and retries in a random time as all servers are not available. The unreliable server can be breakdown when he is working. The breakdown server may not be detected due to some fault matters. Under such a situation, the repair facility will issue a reboot operation to locate the non-detected breakdown server. Once the breakdown server is detected and located, he will be sent for repair. After utilizing the general theory of matrix-geometric approach to determine the stationary distribution, some performance measures are proposed. A single-objective model aiming at minimizing the average cost is constructed. We apply particle swarm optimization algorithm and genetic algorithm to find the optimum combination of parameters. We also formulate a bi-objective model aiming at minimizing the expect cost and the mean waiting time of customers in orbit. Two multi-objective algorithms are developed to solve it.

Optimization of retrial queue with unreliable servers subject to imperfect coverage and reboot delay

It is well known that discrete-time queues are more appropiate than their
continuous-time counterparts for modelling computer and telecommunication
systems. We present a discrete-time Geo/G/1 retrial queue with general retrial
times, Bernoulli feedback and the server subjected to starting failures. We gene-
ralize the previous works in discrete{time retrial queue with unrealiable server
due to starting failures in the sense that we consider general service and ge-
neral retrial times. Also we consider the realistic phenomenon of feedback. We
analyse the Markov chain underlying the considered queueing system and pre-
sent some performance measures of the system in steady-state. We provide two
stochastic decomposition laws. Besides, we prove that the M/G/1 retrial queue
with feedback and starting failures can be approximated by our corresponding
discrete-time system. Some numerical results are given to illustrate the impact
of the unreability and the feedback on the performance of the syste

A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times

The present study deals with the stock-dependent Markovian demand of a retrial queueing system with a single server and multiple server vacation. The items are restocked under a continuous review (s,Q) ordering policy. When there is no item in the system, the server goes on vacation. Further, any arrival demand permits entry into an infinite orbit whenever the server is on vacation. In the Matrix geometric approach with the Neuts-Rao truncation technique, the steady-state joint distribution of the number of customers in orbit, the server status, and the inventory level is obtained. Under the steady-state conditions, some significant system performance measures, including the long-run total cost rate, are derived, and the Laplace-Stieltjes transform is also used to investigate the waiting time distribution. According to various considerations of uncontrollable parameters and costs, the merits of the proposed model, especially the important characteristics of the system with stock dependency over non-stock dependency, are explored. Ultimately, the important facts and ideas behind this model are given in conclusion. 

https://www.mdpi.com/2227-9717/10/1/176/pdf?version=1642491306

Analysis of Stock-Dependent Arrival Process in a Retrial Stochastic Inventory System with Server Vacation

This paper presents a steady-state analysis of an M/M/1 retrial queue with working vacations, in which the server is subject to starting failures. The proposed queueing model is described i...

Performance analysis and optimization of a retrial queue with working vacations and starting failures

This paper presents a steady-state analysis of an M/M/2 queue with heterogeneous servers (Server 1 and Server 2). Server 1 is reliable and may leave for a vacation when the system becomes empty. Se...

Dong Yuh Yang

Papers

Performance analysis and optimization of a retrial queue with working vacations and starting failures

Modelling and optimisation of a two-server queue with multiple vacations and working breakdowns