Ho Woo Lee

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.

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.

DECOMPOSITIONS OF THE QUEUE LENGTH DISTRIBUTIONS IN THE MAP/G/1 QUEUE UNDER MULTIPLE AND SINGLE VACATIONS WITH N-POLICY

In this paper, we establish an explicit form of matrix decompositions for the queue length distributions of the MAP/G/1 queues under multiple and single vacations with N-policy. We show that the vector generating function Y (z) of the queue length at an arbitrary time and X (z) at departures are decomposed into Y (z) = p idle (z)ζ Y (z) and X (z) = p idle (z)ζ X (z) where p idle (z) is the vector generating function of the queue length at an arbitrary epoch at which the server is not in service, and ζ Y (z) and ζ X (z) are unidentified matrix generating functions.

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.

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.

Queue length analysis of MAP/G/1 queue under D-policy

Abstract We study the queue length distribution of a queueing system with MAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths will be derived from these transform results. A numerical example is provided.

BMAP/G/1 Queue Under D-Policy: Queue Length Analysis

ABSTRACT We study the queue length distribution of a queueing system with BMAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths are derived and a numerical example is presented.

Queue length and waiting time analysis of a batch arrival queue with bilevel control

Abstract Scope and Purpose —This paper concerns the analysis of a production system in which an early set-up and N -policy are employed. We model the system by M x /M/1 queue with bilevel control and obtain the queue length and waiting time distributions. In general, with more than two control policies mixed up, obtaining the waiting time distribution is extremely complicated due to the dependencies between the random variables involved. The methodology employed in this paper can be extended to other batch arrival queueing system with more complicated control policies. We consider an M x /M /1 queue with bilevel control. As soon as the system empties, the server becomes idle until m or more customers accumulate, at which time the server starts a set-up. After the set-up, if N or more customers are in the system ( N ⩾ m ) the server immediately start serving the customers in FCFS fashion. If less than N customers are in the system after the set-up, the server remains dormant in the system and wait until N or more customers accumulate. We derive the distributions of the number of customers in the system and the waiting time.

A new approach to the queue length and waiting time of BMAP/G/1 queues

We apply the factorization principle to derive the generating function of the queue length and the vector Laplace-Stieltjes transform of the waiting time of a BMAP/G/1 queue. The mean performance measures are provided with a computational experience.