A Gated Processor - Sharing M / G / 1 Queue with Limited Number of Service Positions

In this note, we consider a single server queue with gated processor-sharing discipline. Customers arrive according to a Poisson process with rate A and service times are independent and identically distributed (i.i.d.) as an arbitrary random variable S with distribution function H(t) and LST H(s). In the gated processor-sharing discipline, when an arriving customer finds the queue empty, he is immediately taken into service and another newly arriving customer is not allowed to enter service while he is receiving service. When he leaves the system, the server admits at most m waiting customers to enter service on firstcome first-served (FCFS) basis and serves this batch by the processor-sharing discipline. Waiting customers are not allowed to enter service while this batch is receiving service. This process of selecting at most m waiting customers for service after the service completion of each batch continues until the queue becomes empty. It is clear that for m = 1, this system reduces to MIG/1 FCFS queue. For any m, the work in this system and the distribution of busy period are identical to those for the MIGII FCFS queue. The traffic intensity, p, is defined by p = AE(S). Furthermore, we assume that p < 1 to ensure the existence of steady state. The processor-sharing discipline are frequently used to model computer systems (see Kleinrock [6]) and communication systems. Recently the gated processor-sharing queue has been analyzed by Rege and Sengupta [8] and AviItzchak and Halfin [1]. In [8], the MIM/1 queue has been analyzed for finite m and m = oo. In [1], the M/G/1 queue has been analyzed for m = oo. In this note, we study the gated processor-sharing M/G/1 queue for finite m. The main purpose of this note is to find the mean waiting time and the conditional mean response time (the mean time in the system by a customer whose service requirement is x). It is well known that the conditjonal mean response time in the ordinary processor-sharing queue is linear in x (e.g., see Heyman and Sobel [5]). However, in the gated processor-sharing queue, it is found that the conditional mean response time is not linear in x. Furthermore, it is found that the gated processor-sharing discipline gives shorter response