An Algorithm for Estimating the Expected Number of Customers for a Class of Markovian Queueing Systems

An algorithm is presented for estimating the expected number of customers for a class of Markovian queueing systems. The class is characterized by those systems whose transition matrix for the underlying customer arrival and departure process is finite, irreducible, and aperiodic. The algorithm does not depend on a closed-form solution for the limiting behavior of the queue. The expected number of customers is frequently used as a measure of effectiveness to describe the behavior of the system or to optimize its design or control. To calculate such a quantity one must usually obtain a closed-form expression for the steady-state probabilities. Unfortunately, of the myriad of Markovian queueing systems, only a few have known closed-form expressions for their steady-state probabilities. The most well-known, using Kendall’s notation, are the M/M/1 and the M/M/c system. The algorithm described below estimates the expected number in the system under steady-state without a need for closed form steady-state probabilities. All that is needed is the transition matrix for the underlying Markov chain.

MATCHED QUEUEING SYSTEMS WITH A DOUBLE INPUT

In this paper, we study the matched queueing system with a double input, MoM/PH/1,where the two inputs are two independent Poisson processes, and the service time is of PH-distribution.The L.S. transforms and the expectations of the distributions of occupation time and virtual waiting time of the type-Ⅰ customer are derived.The probability that the server is working, the mean non-idle period, and the mean busy period are also derived. The related algorithms are given with numerical results.

Approximations for a Queueing Game Model with Join-the-Shortest-Queue Strategy

This paper investigates a partially observable queueing system with N nodes in which each node has a dedicated arrival stream.There is an extra arrival stream to balance the load of the system by routing its customers to the shortest queue.In addition,a reward-cost structure is considered to analyse customers'strategic behaviours.The equilibrium and socially optimal strategies are derived for the partially observable mean field limit model.Then,we show that the strategies obtained from the mean field model are good approximations to the model with finite N nodes.Finally,numerical experiments are provided to compare the equilibrium and socially optimal behaviours,including joining probabilities and social benefits for different system parameters.

A queueing-inventory model to control the congestion of patients and medical waste in the medical centers, a case study

During epidemics,controlling the patients’congestion is a way to reduce disease spreading.Raising medical demands converts hospitals into one of the sources of disease outbreaks.The long patient waiting time in queues to receive medical services leads to more casualties.The rise of patients increases their waste,which is another source of disease outbreak.In this study,a mathematical model is developed to control patients’congestion in a medical center and manage their waste,considering environmental issues.Besides a queueing system controlling the patients’congestion in the treatment center,another queue is considered for vehicles.An inventory model is employed to prevent waste accumulation.The developed model is solved and reaches an exact solution in small size,and obtains an acceptable solution in large size using the Grasshopper algorithm.A case study is considered to demonstrate the model’s applicability.Also,Sensitivity analysis and valuable managerial insights are presented.