A Bulk-Arrival Bulk-Service Queue with a Provision to Allow Late Entries into Certain Batches Being Served

Abstract

      This paper deals with the single server queueing system M^r/M(a,d,b)/1 , in which customers arrive in batches of variable size k (1≤k≤r). Service is also given in batches, but there is a criterion of minimum and maximum number of customers for providing service in a batch.  Minimum number of customers required to initiate service is a≥ 1 and maximum number of customers who may be served in a batch is b≥ a.It is further assumed that if the size of the batchwhich is being served, is less than some fixed integer d (a≤d≤b), the late entries can join the batch being served,until the limit d is reached, without affecting the service time of the ongoing service. Steady-state probabilities are found in all the three cases, when (i) d<a+r, (ii) d=a+r, and (iii) d>a+r, to determine mean queue length. These type of queueing systems are widely found in transportation networks, computer communication networks and traffic control systems to name a few.