Parallel queues with common service process and their input controls

Abstract

In the parallel queue with common service process, there are m independent arrival processes to m waiting spaces. When customers are waiting in all of the waiting areas, the server accepts a customer from each waiting space, and the service is made simultaneously to the m customers. This paper discusses the fundamental properties of such a system, presenting the approximate analysis, qualitative interpretations, and simulation. It has been well known that this kind of system is essentially unstable for any load. This paper presents first the qualitative interpretation for the instability and discusses its relation to the cointossing. Then several heuristic input control schemes are proposed. Their usefulness, control characteristics and the effect of the dalay in the control path are discussed. It is shown as a result that the system can easily be stabilized by a weak input control, and can be stabilized for the case of m =2, even if only one of the queues is controlled. It is indicated also that a too strong control has an adverse effect if there exists a delay in the control path. Several additional observations are made.

Transient and asymptotic behavior of synchronization processes in assembly-like queues

“…Sasieni (1961) also showed that if the arrival and service rates are independent of the queue length, then at least one of the buffers will be unstable and grow indefinitely as time evolves. Through simulation experiments, Fukuda et al (1988) numerically showed that this instability occurs independently of the load of the server. For a multiple input generalization of the GI /G/1 queue, Harrison (1973) proved that an equilibrium of the system never exists if the capacity is infinite.…”

“…Hence, in order to avoid instability, one needs some mechanism of controlling the buffer contents in the system by limiting inputs. Simple control limit policies have been discussed in the previous literature (Duenyas and Hopp 1993;Foster 1959;Fukuda et al 1988;Keblis and Duenyas 1999). Kashyap (1966) assumed limited waiting spaces for both customers as a mechanism to limit the buffer capacity.…”