We present a new approach to derive joint distributions for stationary Poisson loss systems. In particular, for M/M/m/0 and M/M/1/n we find the Laplace transforms (with respect to time t) of the probability that at time t there are i customers in the system and during [0,t], j customers are refused admissions for M/M/m/0 we further determine the LT of the probability that the system was full for less than s time units during [0,t] and serves i customers at time t. Explicit formulas for the corresponding moments are also given.

Introduction to the Theory of Queues.

Generating function analysis of some joint distributions for Poisson loss systems

Abstract By applying the Takács’ technique about the busy period to the regenerative cycle method, this paper gives the strict proof for the time average distributions of the queue length in M/G/1 without depending on other methods. Moreover we extend its proof from the service time to the completion time(CT). That is, we choose the stochastic behavior on the completion time as the regenerative cycle and, by using its PGF, represent the queue length distributions in M/CT/1, M/CT/1 with N -policy, M/CT/1 with multiple vacation and their combinations. Our completion time is able to contain the additional service time, the vacation, the loss interval and the batch arrival. We can also consider some service disciplines on it like time-controlled service discipline. Thus the completion time realizes the wider application of the regenerative cycle method, unifies various variants of the fundamental models and derives their probability generating functions.

background or methods

QUEUE LENGTH DISTRIBUTION IN M/G/1, M^x/G/1 AND THEIR VARIANTS WITH COMPLETION TIME

Smart On-Street Parking System to Predict Parking Occupancy and Provide a Routing Strategy Using Cloud-Based Analytics

Transient behavior of multi-server queues with recurrent input and exponential service times

In this paper we consider the analysis of an M/G/1 queue with working vacation. In contrast to the previous literature where the working vacation starts when all customers are served (exhaustive discipline) we consider the case where the vacation period starts when the customers present at the system at beginning of the service period are served (gated discipline). The analysis of the model with gated discipline requires a different approach than the one with exhaustive discipline.

We present the probability-generating function of the number of customers in the system and the Laplace-Stieljes transform of the stationary waiting time.

Additional excerpts

/pdf/m-g-1-queue-with-exponential-working-vacation-and-gated-4gavwz8wtf.pdf

Introduction to the Theory of Queues.

Citations

Generating function analysis of some joint distributions for Poisson loss systems

QUEUE LENGTH DISTRIBUTION IN M/G/1, M^x/G/1 AND THEIR VARIANTS WITH COMPLETION TIME

Smart On-Street Parking System to Predict Parking Occupancy and Provide a Routing Strategy Using Cloud-Based Analytics

Transient behavior of multi-server queues with recurrent input and exponential service times

M/G/1 queue with exponential working vacation and gated service

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