Sojourn times in a multiclass processor sharing queue

We consider a processor sharing queue with several customer classes. For an arbitrary customer of class i we show that the sojourn time distribution is regularly varying of index -\nu_i iff the service time distribution is regularly varying of index -\nu_i, and derive an explicit asymptotic formula. Furthermore, the tail of the sojourn time distribution of customer class i is shown to be unaffected by the tails of the service time distributions of other customer classes, even if some of the latter tails are heavier. This result implies that, when the sojourn time of a customer is large, this is not due to long service requirements of other customer types. In particular, short-range dependent traffic does not suffer from longe-range dependent traffic if processor sharing is used as a service discipline

A fluid queue with a finite buffer and subexponential input

We consider a fluid model similar to that of Kella and Whitt [33], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established in [33] are extended to the finite buffer case. It is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N on-off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Waiting-time asymptotics for the M/G/2 queue with heterogeneous servers

This paper considers a heterogeneous M/G/2 queue. The service times at server 1 are exponentially distributed, and at server 2 they have a general distribution B(.). We present an exact analysis of the queue length and waiting time distribution in case B(·) has a rational Laplace-Stieltjes transform. When B(·) is regularly varying at infinity of index -\nu, we determine the tail behaviour of the waiting time distribution. This tail is shown to be semiexponential if the arrival rate is lower than the service rate of the exponential server, and regularly varying at infinity of index 1 - \nu if the arrival rate is higher than that service rate