Kyung C. Chae

Batch arrival queue with N -policy and single vacation

Abstract We consider an M x / G /1 queueing system with N -policy and single vacation. As soon as the system becomes empty, the server leaves the system for a vacation of random length V . When he returns from the vacation, if the system size is greater than or equal to predetermined value N (threshold), he beings to serve the customers. If not, the server waits in the system until the system size reaches or exceeds N . We derive the system size distribution and show that the system size distribution decomposes into two random variables one of which is the system size of ordinary M x / G /1 queue. The interpretation of the other random variable will also be provided. We also derive the queue waiting time distribution of an arbitrary customer. Finally we develop a procedure to find the optimal stationary operating policy under a linear cost structure.

Transform-free analysis of the GI/G/1/K queue through the decomposed Little's formula

In this paper, we consider the steady-state queue length distribution of the GI/G/1/K queue. As a result, we obtain transform-free expressions for the steady-state queue length distributions at an arrival, at a departure and at an arbitrary time, all in product forms. The results are obtained by what we call the decomposed Littles formula, which applies the Littles formula L = λW to the nth waiting position in the queue. Utilizing the results, we improve and generalize existing bounds on the difference between the time average and arrival (departure) average mean queue lengths, and propose a two-moment approximation for the queue length. To evaluate the approximation, we focus on the probability of customer loss and the mean queue length, which are of great practical importance. Our approximations turn out to be remarkably simple yet fairly good especially in heavy traffic.

Discrete-time queues with discretionary priorities

In this paper, we consider a discrete-time two-class discretionary priority queueing model with generally distributed service times and per slot i.i.d. structured inputs in which preemptions are allowed only when the elapsed service time of a lower-class customer being served does not exceed a certain threshold. As the preemption mode of the discretionary priority discipline, we consider the Preemptive Resume, Preemptive Repeat Different, and Preemptive Repeat Identical modes. We derive the Probability Generating Functions (PGFs) and first moments of queue lengths of each class in this model for all the three preemption modes in a unified manner. The obtained results include all the previous works on discrete-time priority queueing models with general service times and structured inputs as their special cases. A numerical example shows that, using the discretionary priority discipline, we can more subtly adjust the system performances than is possible using either the pure non-preemptive or the preemptive priority disciplines.

The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV). During the SWV, the server works at a different rate rather than completely stopping working. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrary customer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it with the GI/M/1/SWV queue.

The Geo/G/1 Queue with Disasters and Multiple Working Vacations

In this paper, we first consider a Geo/G/1 queue with disasters that remove all workloads from the system upon their occurrence. We present the steady-state queue-length distribution of the Geo/G/1 queue with disasters. Using this result, we then analyze the Geo/G/1 queue with multiple working vacations in which the server works at a different rate rather than completely stopping during the vacation period. We also present the steady-state queue-length distribution of the Geo/G/1 queue with multiple working vacations.

ANALYSIS OF M/G/1 STOCHASTIC CLEARING SYSTEMS

In this paper, we consider M/G/1 queuing systems governed by a stochastic clearing mechanism, called “disaster,” which removes all workload in the system whenever it occurs to the system. The clearing mechanism of disasters can be applied to computer systems in the presence of a virus as a clearing operation of all stored messages present in the system. We present the system size distribution and the sojourn time distribution.

Workload and Waiting Time Analyses of MAP/G/1 Queue under D-policy

We study the workload (unfinished work) and the waiting time of the queueing system with MAP arrivals under D-policy. The D-policy stipulates that the idle server begin to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We first set up the system equations for workload and obtain the steady-state distributions of workloads at an arbitrary idle and busy points of time. We then proceed to obtain the waiting time distribution of an arbitrary customer based on the workload results. The M/G/1/D-policy queue will be investigated as a special case.

A Two-Moment Approximation for the GI/G/c Queue with Finite Capacity

In this paper, we consider the steady-state queue length of the multiserver finite-capacity GI/G/c/c+r queue. As a result, we first obtain an exact transform-free expression for the steady-state queue-length distribution. Making use of this result, we then present a simple two-moment approximation for the queue-length distribution. From this, approximations for some important performance measures, such as the loss probability, the mean queue length, and the mean waiting time, are also obtained. In addition, we propose an approximation for the minimal buffer size that keeps the loss probability below an acceptable level. Extensive numerical experiments show that our approximation is extremely simple yet fairly good in its performance.

An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

For a broad class of discrete- and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some multiserver queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the probability generating functions (PGFs) of the stationary numbers in queue and in system, as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with batch Markovian arrival process (BMAP) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Maintenance of deteriorating single server queues with random shocks

We consider the maintenance of single server queues in which the deterioration of a server is subject to random shocks. Shock arrivals deteriorate the server by a random amount. A maintenance policy is proposed whereby the server is repaired whenever its state is above a specified maintenance level. We present the system size distribution and sojourn time distribution. We derive the long run average cost, considering holding cost and repair cost. We analyze the proposed maintenance policy based on the cost analysis.