Kuo-Hsiung Wang

A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

Abstract We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.

Optimal operation of a markovian queueing system with a removable and non-reliable server

Abstract This paper deals with the optimal operation of a single removable and non-reliable server in a Markovian queueing system under steady-state conditions. The server can be turned on at arrival epochs or off at departure epochs. We assume that the server may break down only if working and requires repair at a repair facility. Interarrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to the system: a holding cost for each customer in the system per unit time, costs per unit time for keeping the server on or off, a breakdown cost per unit time when a server fails, and fixed costs for turning the server on or off. The total expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.

Control Policies of an M/G/1 Queueing System with a Removable and Non‐reliable Server

This paper considers a single non-reliable server in the ordinary M/G/1 queueing system whose arrivals form a Poisson process and service times are generally distributed. We also study a single removable and non-reliable server in the controllable M/G/1 queueing systems operating under the N policy, the T policy and the Min(N, T) policy. It is assumed that the server breaks down according to a Poisson process and the repair time has a general distribution. In three control policies, we show that the probability that the server is busy in the steady-state is equal to the traffic intensity. It is shown that the optimal N policy and the optimal Min(N, T) policy are always superior to the optimal T policy. Sensitivity analysis is also investigated.

A recursive method for the N policy G/M/1 queueing system with finite capacity

Abstract This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

Abstract We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.

Comparative analysis of a randomized N-policy queue: An improved maximum entropy method

We analyze a single removable and unreliable server in an M/G/1 queueing system operating under the -policy. As soon as the system size is greater than N, turn the server on with probability p and leave the server off with probability (1-p). All arriving customers demand the first essential service, where only some of them demand the second optional service. He needs a startup time before providing first essential service until there are no customers in the system. The server is subject to break down according to a Poisson process and his repair time obeys a general distribution. In this queueing system, the steady-state probabilities cannot be derived explicitly. Thus, we employ an improved maximum entropy method with several well-known constraints to estimate the probability distributions of system size and the expected waiting time in the system. By a comparative analysis between the exact and approximate results, we may demonstrate that the improved maximum entropy method is accurate enough for practical purpose, and it is a useful method for solving complex queueing systems.

Comparative Analysis for the N Policy M/G/1 Queueing System with a Removable and Unreliable Server

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E2(E3)/1, M/H2(H3)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.