Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.