Delay Distributions in Discrete Time Multiclass Tandem Communication Network Models

An exact computational algorithm for the solution of a discrete time multiclass tandem network with a primary class and cross-traffic at each queue is developed. A sequence of truncated Lindley recursions is defined at each queue relating the delays experienced by the first packet from consecutive batches of a class at that queue. Using this sequence of recursions, a convolve-and-sweep algorithm is developed to compute the stationary distributions of the delay and inter-departure processes of each class at a queue, delays experienced by a typical packet from the primary class along its path as well as the mean end-to-end delay of such a packet. The proposed approach is designed to handle the non-renewal arrival processes arising in the network. The algorithmic solution is implemented as an abstract class which permits its easy adaptation to analyze different network configurations and sizes. The delays of a packet at different queues are shown to be associated random variables from which it follows that the variance of total delay is lower bounded by the sum of variances of delays at the queues along the path. The developed algorithm and the proposed lower bound on the variance of total delay are validated against simulation for a tandem network of two queues with three classes under different batch size distributions

Some algorithms for discrete time queues with finite capacity

We consider a discrete time queue with finite capacity and i.i.d. and Markov modulated arrivals, Efficient algorithms are developed to calculate the moments and the distributions of the first time to overflow and the regeneration length, Results are extended to the multiserver queue. Some illustrative numerical examples are provided

Asymptotics for transient and stationary probabilities for finite and infinite buffer discrete time queues

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buffer length m. Let $T_m$ be the first time an overflow occurs. We obtain asymptotic rate of growth of moments and distributions of $T_m$ as $m\rightarrow \propto$. We also show that under general conditions, the overflow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overflow epochs are qualitatively as well as quantitatively different from the excursion process of an infinite buffer queue studied in continuous time in the literature. Asymptotic results for several other characteristics of the loss process are also studied, e.g., exponential decay of the probability of no loss (for a fixed buffer length) in time [0, n], as n $\rightarrow \propto$, total number of packets lost in [0, n], maximum run of loss states in [0, n]. We also study tails of stationary distributions. All results extend to the multiserver case and most to a Markov modulated arrival process