ON THE STRUCTURE OF AN INSENSITIVE GENERALIZED SEMI-MARKOV PROCESS WITH REALLOCATION AND POINT-PROCESS INPUT

A generalized semi-Markov process with reallocation (RGSMP) was introd uced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-M arkov process (GSMP). For stationary RGSMP whose initial distribution has a product Form, we show that, for a randomly chosen clock of a fix ed insensitive type, if the lifetime of this clock is changed to infin ity, then the background process is stationary under a certain time ch ange. This implies that the expected time required for the tagged cloc k to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity resu lts are known for two special RGSMPs: ordinary GSMP and Kelly's symmet ric queue. Our results not only extend them to a general RGSMP but als o give more detailed formulas, which allow us to calculate for instanc e the expected attained sojourn time while the background process is i n a given state. Furthermore, we remark that analogous results hold fo r GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).