Finite Queues and Cyclic Queues

Abstract

The finite queue problem, for which tables exist [Peck, L. G., R. N. Hazelwood. 1958. Finite Queuing Tables . ORSA Publications in Operations Research No. 2. Wiley, New York.], is a special case of the cyclic queue [Koenigsberg, E. 1958. Oper. Res. Quart. 9 22–35]. We consider a closed system with two stages, the first a repair stage and the second an operating stage. There are N machines in the system, of which a maximum of A can be operating or productive at any one time ( A operators). When breakdowns occur at a mean rate μ 2 the machines enter the repair stage which has M parallel servers ( M repairmen), who service the machines at a mean rate μ 1 . If μ 1 and μ 2 are defined by an exponential distribution, then the numbers N , A , M , and μ 2 /μ 1 define the output of the system. When A = N the problem is identical to the Swedish Machine problem for which tables are already available [Peck, L. G., R. N. Hazelwood. 1958. Finite Queuing Tables . ORSA Publications in Operations Research No. 2. Wiley, New York.].