A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution

For the GI/G/1 queueing model with traffic load a<1, service time distribution B(t) and interarrival time distribution A(t), whenever for t→∞1−B(t)∼c(t/β)ν


Introduction
For the GI/G/1 queue, denote by A(t) and B(t) the interarrival time distribution and service time distribution, respectively, and by a the traffic load, with a < 1.The distribution B(t) is said to have a Pareto-type tail if: for tc, 1Work carried out under project LRD.
w shall denote a stochastic variable with distribution W(t), the stationary distribu- tion of the actual waiting time of the GI/G/1 model.Write A. [ .l a F(U)sin(v -1)rJ u i a cr ariable with distribution W(t), the stationary distribu- tion of the actual waiting time of the GI/G/1 model.Write A. [ .l a F(U)sin(v -1)rJ u i a cr

(1.2) h (1.2) here r(. is the gamma function and xa, a real, is defined by its principal value, i.e.,. by its principal value, i.e.,.

it is positive for x positive.

Theorem: When B(t) has a Pareto-type tail as specified in (1.1) and when t"dA(t)< (x) for a # > u, (

then the stochastic variable (1-a) u-lw/fl converges for aT1 in it is positive for x positive.
The theorem stated above is a heavy traffic result.The classical heavy traffic theorem for the GI/G/1 model, cf. [3],Section III.7.2, requires the finiteness of the second moment of A(t) and that of B(t).In a forthcoming paper by O.J. Boxma and the present author, generalizations of the theorem above will be discussed.

Proof of the Theorem
We consider first the case with all c n -O, n-1,...,N.Consequently, it is seen from (1.1) that we may write: for t > c ). + s(t), (p)" j e-ptdB(t), By using (2.2) it is readily seen that gl(P) is a regular function of p for Re p > -5.
Let be the idle period, i.e., the difference of a busy cycle and the busy period contained in this busy cycle.The relation between the distributions of w and is 0with n the number of customers served in a busy cycle.With cr(2.17