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

Abstract

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distributionW(x) has an asymptotically exponential tail, i.e., 1 –W(x) ∽Ke–ckx. The parameter k is the unique positive number satisfyingT*(ck)S*(–k) = 1, whereT*(s) andS*(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 constantKis not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.