The Virtual Waiting Time of the M/G/1 Queue with Impatient Customers

The M/G/1 queue with impatient customers is studied. The complete formula of the limiting distribution of the virtual waiting time is derived explicitly. The expected busy period of the queue is also obtained by using a martingale argument.     

Approximate sojourn time distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system scheduled by the discriminatory processor sharing discipline. Poisson arrivals of customers and exponential amounts of service requirements are assumed. At any moment of being served, a customer can leave the system without completion of its service. In the asymptotic regime, where the ratio of the time scales of the two-class customers is infinite, we obtain the conditional sojourn time distribution of each class customers. Numerical experiments show that the time scale decomposition approach developed in this paper gives a good approximation to the conditional sojourn time distribution as well as the expectation of it.     

Busy Periods of Poisson Arrival Queues with Loss

We consider two queues with loss, one is the finite dam with Poisson arrivals and the other is the M/G/1 queue with impatient customers. We use the method of Kolmogorov's backward differential equation and construct a type of renewal equation to obtain the Laplace transform of busy(or wet) period in both queues. As a consequence, we provide the explicit forms of expected busy periods.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

The stationary workload of the <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 queue with impatient customers

We consider a G/M/1 queue in which the patience time of the customers is constant. The stationary distribution of the workload of the server, or the virtual waiting time, is derived by the level crossing argument. To this end, we obtain the expected downcrossings of a level in the workload process during a busy cycle and then the expected length of a busy cycle. For both the expectations, we use the dual property between the M/G/1 and G/M/1 queue.