Exact distribution of discrete-time D-BMAP/G/\inf queueing model

TL;DR

Addresses the exact time-dependent distribution of the number of customers in a discrete-time D-BMAP/G/infty queue. The authors construct an effective Markovian dynamics using the auxiliary process N(s;t) and derive a generating-function recursion with T(z,s;t) = D(Phi(t-s) z + 1 - Phi(t-s)); this yields explicit expressions for $p_m(t)$ and factorial moments. For the discrete-time M/M/infty case, they obtain a closed form $G(z,t) = product_{i=1}^t (1 + p alpha^i (z-1))$ with $m(t) = p alpha (1 - alpha^t)/(1 - alpha)$ and $\sigma^2(t) = p alpha (1 - alpha^t)/(1 - alpha) - p^2 alpha^2 (1 - alpha^{2t})/(1 - alpha^2)$, and show a sub-Poisson Fano factor. The framework enables exact performance analysis of discrete-time infinite-server queues with general arrivals and service distributions and bridges discrete-time and continuous-time limits.

Abstract

In this paper, we consider discrete-time D-BMAP/G/\inf queueing model. We construct effective discrete-time Markovian dynamics for this model and utilize it to derive exact time-dependent distribution of customer number and the corresponding moments for the original queueing model. Numerical simulations are used to verify our results. Using our result, we provide analytical distribution for discrete-time M/M/\inf, and then compare it with the distribution of continuous-time M/M/\inf.

Figures (2)

• Figure 1: Illustration of the effective process $N(s;t)$ for D-BMAP/G/$\infty$. Here the blue curve is a trajectory of $N(t)$, and the other three curves are the trajectories of $N(s;t)$ with different $t$. Here the D-BMAP is given by $D_0 = [0.1, 0.2; 0.05, 0.2], D_1 = [0.1, 0.1; 0.1, 0.1], D_2 = [0.1, 0.15; 0.1, 0.1], D_3 = [0.05, 0.1; 0.1, 0.05], D_4 = [0.05, 0.05; 0.1, 0.1]$, and the service times are Poissonian distributed with mean value being $4$.
• Figure 2: Comparison between exact and numerical results for a two-state D-BMAP/G/$\infty$ queueing model given by Eqs. \ref{['example1']} and \ref{['ex1']}. Here (a) compares the mean customer numbers, (b) compares the second-order factorial moments, and (c) compares the exact time-dependent distributions. The numerical results is obtained by averaging over 50000 realizations. The details of the SSA for D-BMAP/G/$\infty$ can be found in Appendix \ref{['ssa']}.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 5
• Theorem 1