Transient Behavior of the Kendall Birth-Death Process - Applications to Capacity Expansion for Special Services

In this article, the transient behavior of the Kendall birth-death process* with immigration is examined, and some applications of the * The Kendall birth-death process is one in which the transition rates are proportional to the state. 57 process to capacity expansion problems are discussed. The choice of such a process was motivated by the search for a model for special services point-to-point circuit demand, a model which would be used as a tool for determining facility network circuit routing strategies. Special services demand generally consists of demand for full-time dedicated circuits (e.g., foreign exchange lines, WATS lines, data lines), as opposed to the message-traffic offered load which consists of demand for the use of common facilities for a relatively short period of time. Thus, the system examined is characterized by the stochastic process JAt) with realizations (states) n = 0, 1, · · ·, oo, where n might refer to the number of working circuits or some other facility, rather than to the number of busy trunks, as in the message-traffic case. By definition, the birth-death process1 allows transitions from some state n to n + 1 via a birth (circuit connect), or to n -- 1 via a death (circuit disconnnect). The transition rates are A,, for the births and in for the deaths, both of which are chosen proportional to n for the following reasons. It is clear, for special services, that the rate of disconnects, /*,,, is state dependent. There are, in fact, indications2 that in is a monotonically increasing function of n.