WAITING-TIME APPROXIMATIONS FOR CYCLIC-SERVICE SYSTEMS WITH MIXED SERVICE STRATEGIES

An approximation method for mean waiting times in cyclic-service systems with a mixture of exhaustive/ gated/ I-limited service strategies is presented. The method provides a unification and generalization of some known good approximations for single strategies. The results of an exact analysis of the two-queue case where one queue is served exhaus­ tively and the other queue I-limited, support the approximation idea. Token-passing protocols are becoming increasingly popular for application in Local Area Networks (LAN's) with a ring or bus topology. With these protocols, rather than to attach all stations to a sin­ gle subnetwork such as a ring, for various reasons it is preferred to interconnect several subnetworks. When there is no need for protocol conversion in the interconnection a bridge is used: a dedicated station in a LAN providing interconnection between subnetworks, at a very low level of architecture. When a LAN is to be connected to a network of an other type, a gateway must be used: from the point of view we shall adopt here, a gateway is a dedicated station in a LAN providing the intercon­ nection between networks, where there may be a need for protocol conversion. Of course a gateway has to be substantially more complicated and operates at a higher level of architecture than a bridge does, and hence is usually much slower compared to a bridge. So, essentially, we distinguish between three types of stations in a token-passing (sub-)network: ordinary stations, bridge stations, and gate­ way stations. It is clear that, since these types of stations each have different characteristics, it may be advantageous to assign them different priorities with respect to the communication protocol. In general, the performance of polling schemes, of which the token-passing protocol is an example, can be analyzed by studying single-server, multi-queue queueing systems. For example in a token ring LAN the common transmission medium may be represented by the single server, and the worksta­ tions attached to the ring by the queues. The circulation of the token along the ring implies that the stations are polled in a cyclic order. This paper is concerned with the waiting-time process at the various queues of a polling system as described above. Let us first present a more detailed model description. We consider a system of N queues, Q 1 , ••• ,QN, served sequentially in cyclic order by a single server S. Messages arrive at all queues according to independent Poisson processes with arrival intensities A},A2' ... ,AN. The switch-over times of the server between the i-th and (i + l)-th queue are independent, identically dis­ tributed random variables, with first moment Si, second moment S~2) and Laplace Stieltjes Transform (LSn 0';(.). The first moment of the total switch-over time during a cycle of the server is denoted by s, its second moment by S(2). The service times of type-i messages (messages enqueued in Q;) are independent, identically distributed random variables with first moment Pi and second moment pF). We assume that the arrival process, the service process and the switch-over process are mutually independent. The offered traffic at Q;, Pi, and the total offered traffic, p, are defined as: Pi: ='A;/3;, i=l, ... ,N; P:=Pl+P2+ ... +PN. For the service strategies at the queues we consider three possibilities, which differ in the number of messages which may be served in a queue during a visit of server S to that queue. Assume that S visits Qi. When Qi is empty, S immediately begins to switch to Q; + 1. Otherwise S acts as follows, depending on the service strategy at Q;: