Flow Time in theMAP/G/1Queue with Customer Batching and Setup Times

Abstract

This paper studies flow times in a single server queue with customer batching and setup times. Customers arrive according to a Markov arrival process where every N consecutive customers are gathered into a batch before service. The service time of a batch consists of a setup time followed by N individual processing times. The customer flow time, an important measure in production management, is defined as the time between the arrival of an arbitrary customer and the time its batch leaves the system. The main result of this paper is the Laplace Stieltjes transform of the customer flow time, from which the mean and variance of the customer flow time are derived. An extension of existing results for the MAP/G/1 queue is used to develop an algorithm for computing the mean and an approximation to the variance of the customer flow time. The relationship between the burstiness of the input process and the mean flow time, flow time variance and the corresponding optimal batch sizes is explored with numerical examples