Analysis of Functional Equations in M/G/1 Queueing-System

This chapter summarizes some functional equations which are analyzed by using the lemma of Tak \(\grave{a}\) cs (Method I), Kuczma’s iteration (Method II) and the boundary value method (Method III) [1, 2]. All of functional equations are formulated from the following queueing models: (i) Flexible priority queueing model with controllable parameters. (ii) Alternating service model with gated and Bernoulli services. (iii) Cyclic-polling model with 1-limited service. (iv) Two-stage tandem queueing model attended by a moving single server. These models have been motivated from computer systems in communication networks and telecommunication networks. From the above models, simultaneous functional relations are derived, where the functional relation contains the unknown probability generating functions (PGFs) on stationary queue-length distributions. In these functional relations, the determination of unknown probabilities and PGFs is a main subject, where unknown probabilities can be determined by using Method I and all of unknown PGFs can be determined by using Method II. Further, a new type of functional equations are formulated, a part of which belongs to the so-called open problem regarding the queueing models (ii) and (iii), where model (iii) is concerned with Method III. For the tandem queueing model (iv), we analyze a sojourn time elapsed from message arrival at the first and second stage queues to its service completion at a service-counter. For Method II, the complicated decent-order procedure is necessary for analyses of model (i). Finally, the above analyses of functional equations are basic and applicable to calculations of waiting time and sojourn time which are useful for deriving performance measures.