On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A_n}_n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS_nare i.i.d. exponentially distributed. (SinceA_n=0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].