Kyung-Chul Chae

Analysis of the M x/G/1 queue by N-policy and multiple vacations

We consider an MX/G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary MX/G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure. SYSTEM SIZE; QUEUE WAITING TIME; OPTIMAL OPERATING POLICY AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60K25

Operating characteristics of MX/G/1 queue with N-policy

We consider aMX/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theMX/G/1 queueing system withoutN-policy and the other one has the probability generating function ∑j=0N=1 πjzj/∑j=0N=1 πj, in which πj is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.

Estimating Parameters of the Power Law Process With Two Measures of Failure Time

We introduce a method of combining two time indices, such as mileage and age, into a single index that resembles the Cobb-Douglas production function. Then we present the power law process with this synthesized time index as a model for the reliability ..

A FIXED-SIZE BATCH SERVICE QUEUE WITH VACATIONS

The paper deals with batch service queues with vacations in which customers arrive according to a Poisson process. Decomposition method is used to derive the queue length distributions both for single and multiple vacation cases. The authors look at other decomposition techniques and discuss some related open problems.

A random review replacement model for a system subject to compound poisson shocks

We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply

Presenting the negative hypergeometric distribution to the introductory statistics courses

In this paper, we suggest teaching the negative hypergeometric distribution in introductory statistics courses so as to complete the matching partnerships among the major discrete probability distributions.

Server unavailability reduces mean waiting time in some batch service queuing systems

Abstract This article shows that in some queuing systems server unavailability is sometimes beneficial both to the system owner and to the customers. To be more specific, mean queue size and mean waiting time may decrease even in the presence of server vacations. To this end, we first derive the probability generating function of the batch service M/GB/1 system with multiple vacations. We, then consider the case of exponential service and vacation times and show that under some parameter combinations, mean queue size decreases even in the presence of server vacations. We discuss the implications of this phenomenon.

On the random sum with a stopping rule SN > T>

We consider the random sum Sn >+ X>1 + X>2 + ...+ Xn >with a stopping rule N>= min{n> : Sn > t}>where T>is a predetermined threshold. We assume that X>1 ,X>2 ,...>are independent and identically distributed random variables having positive integer values. N>is also a positive integer‐valued random variable but dependent on the sequence X>1 , X>2 ,...>due to the stopping rule. Given T>and the probability distribution of X,>we first derive the probability distributions of N, Sn,>and of N>conditional on Sn.>Then we compute the first two moments of these three taking an instructive approach based on the absorbing Markov chain.

On the geometric sum of iid random variables

We consider the random sum Sn= X1 + X2+ ... + XNwhere X1, X2,...are independent and identically distributed random variables, AT is a geometric random variable, and Nis in general dependent on X.A common mistake in the literature is corrected.

A resolution of the ‘exchange paradox’

We extend existing resolutions of the exchange paradox. We also clarify issues under debate by correcting some mistakes in the literature.