6 research outputs found
Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals
In this paper, we consider the occupancy distribution for an open network of
infinite server queues with multivariate batch arrivals following a
non-homogeneous Poisson process, and general service time distributions. We
derive a probability generating function for the transient occupancy
distribution of the network, and prove that it is necessary and sufficient for
ergodicity that the expected occupancy time for each batch be finite. Further,
we recover recurrence relations for the transient probability mass function
formulated in terms of a distribution obtained by compounding the batch size
with a multinomial distribution
Non-Stationary Queues with Batch Arrivals
Motivated by applications that involve setting proper staffing levels for
multi-server queueing systems with batch arrivals, we present a thorough study
of the queue-length process , departure process , and the workload process associated with the
M/G/ queueing system, where arrivals occur in
batches, with the batch size distribution varying with time. Notably, we first
show that both and are equal in distribution to an infinite sum
of independent, scaled Poisson random variables. When the batch size
distribution has finite support, this sum becomes finite as well. We then
derive the finite-dimensional distributions of both the queue-length process
and the departure process, and we use these results to show that these
finite-dimensional distributions converge weakly under a certain scaling
regime, where the finite-dimensional distributions of the queue-length process
converge weakly to a shot-noise process driven by a non-homogeneous Poisson
process. Next, we derive an expression for the joint Laplace-Stieltjes
transform of , , and , and we show that these three random
variables, under the same scaling regime, converge weakly, where the limit
associated with the workload process corresponds to another Poisson-driven
shot-noise process