Delays for Last-Come First-Served Service and the Busy Period

Delays for last-come first-served service were first considered by Vaulot 1 for a system with Poisson input to a group of fully accessible servers, each with exponential service time. For this order of service, an arrival which is not served immediately, following the biblical edict, goes to the head of the waiting line. Its consideration has a natural theoretical interest, because, as the opposite of first-come first-served, it seems to be a bound for the gamut of possible service assignments, or at least of those with simple structure. Indeed Vaulot 1 has used it to find the envelope of delay functions for all service assignments. Very briefly, Vaulot's formulation is as follows. Let v,,(t) be the probability that a waiting demand for service, which at a given moment has just become n -f- 1 in line, waits at least t [',,(/) is the complement of a distribution function; v0(t) is the complement of the conditional delay distribution function]. Then, if a is the arrival rate, and b the service rate for each of the c servers, the set of differential recurrence relations for the i'n(t) is vH'(t) = bevn-xit) - (a + bc)vn(t) + av,,+i(t), n = 0,1,