Transient analytical solution of M/D/1/N queues

Abstract

An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.

Modeling queues with simulation versus M/M/C models

“…Two basic approaches are proposed in the literature. The first approach is to explicitly model the system transience and state transition over time as suggested by Abate & Whitt (1987), Lee & Roth (1993), Van Den Berg & Groenendijk (1991, Wang (1999), and Garcia et al (2002). The second approach is to divide time into segments, estimate the performance in each segment using stationary queuing models such as the M/M/C models, and finally average the performance across all segments.…”

Traffic Modelling and Performance Evaluation in the Kotor Cruise Port

“…Hence, the corresponding queuing model in Kendall's notation is M/D/1/m. In [14], the transient analytical solution of M/D/1/m queues based on M/D/1/m as a finite capacity queuing system, with m -1 waiting spaces in the queue is defined. In our study, we also suppose that cruise ships arrive according to a Poisson process at arrival rate λ.…”

“…denote the probability of k arrivals during a service of the ship. Using a probability transition matrix of the embedded Markov chain [14] and [15]; it is possible to derive the stationary probability distribution of the number of cruise ships in the M/D/1/m queue:…”

“…According to the second formula of Theorem 2 from[14], for an average waiting time in the M/D/1/m queue, 's law it follows that the average number of cruise ships in the queue is…”

A new formula for the transient solution of the Erlang queueing model

“…Some interesting work has recently in been done this area by Chaudhry and Zhao (1999), Garcia et al (2002), Jaiswal (1980) and Mederer (2003). Tikhomenko (2005) studied stationary distribution for the Erlang model.…”