Approximations for Product Departures from a Single-Server Station with Batch Processing in Multi-Product Queues

In this paper we consider a single-server station processing jobs belonging to multiple product classes. The processing at the station is in batches of fixed size, r. Job arrivals in different product classes are independent of the arrivals in other classes. The arrivals within each product class and the service times are assumed to have general, independent and identical distributions. Based on approximate analyses, we present an estimate for the mean number of jobs and provide two complementary characterizations of the product departure streams. We present methods to compute the squared coefficient of variation of the departure intervals and the probability distribution of the lot sizes of product departures. The computational results reported in this paper demonstrate that the accuracy of the approximations is acceptable in most applications. Based on these results, we identify conditions under which the estimates can be expected to perform well. The methods developed in this paper complement the decomposition approach for open queueing networks proposed by Shanthikumar and Buzacott (Shanthikumar, J. G., J. A. Buzacott. 1981. Open queueing network models of job shops. Internat. J. Production Res. 19(3) 255--266.) and Whitt (Whitt, W. 1983. The queueing network analyzer. Bell System Tech. J. 62(9) 2779--2815.) and permit analysis of networks with some types of batch processing.