SMVA: A Stable Mean Value Analysis Algorithm for Closed Systems with Load-Dependent Queues

Abstract

The load-dependent Mean Value Analysis (MVA) algorithm suffers from numerical instability issues. Different techniques have been adopted to avoid these issues, however, they have either complexity problems or restrictive assumptions. In this paper, we introduce a numerically Stable MVA (SMVA) algorithm for closed product-form queueing networks that allows for load-dependent queues. The SMVA algorithm is inspired by Seidmann’s approximation for the numerical stability, and employs the Bard-Schweitzer approximation for the accuracy. The SMVA algorithm offers a numerically stable, efficient, and accurate approximate solution. We validate SMVA by comparing it to other MVA algorithms in concrete examples, and analyse its errors. We also extend it to a multi-class model.

Appendix

Algorithm 2 The single-class A-SMVA algorithm

In Algorithm 1, we estimate the mean number of jobs at a delay centre, $Q_m^e(n)$. In the same iteration, new values are calculated as $Q_m^o(n)$. A natural thought would be to add a comparison between these two values, similar to a technique in the Bard-Schweitzer approximation. To accomplish this, we propose an alternative SMVA algorithm (A-SMVA) for a single-class system. The details of A-SMVA can be seen in Algorithm 2. Compared to SMVA, A-SMVA differs as follows:

• Iterations over n = 1 → N are removed. Instead, we focus only on the performance metrics with N jobs in the system.

• Initialize $Q_m^e(n)$ with estimated values, e.g., N∕(M + 1).

• Employ (2.4) to replace $Q^a_m(N-1)$ by $Q^q_m(N)$ in (2.3), which is $Q^q_m(N)\times (N-1)/N$.

• Choose an error criterion—𝜖, e.g., 0.01.

• Compare the difference between $Q_m^e(n)$ and $Q_m^o(n)$, and compare the difference between $Q^q_m(N)$ and $Q^a_m(N)$. If the maximum difference is larger than 𝜖, replace $Q_m^e(n)$ by $Q_m^o(n)$, and $Q^q_m(N)$ by $Q^a_m(N)$. Otherwise, stop the iteration.

We set 𝜖 = 0.01, and compare the results of A-SMVA with the results of SMVA with the same input parameters in Sect. 2.5.1. The estimated mean response times from A-SMVA are slightly larger than the results from SMVA, but they have very similar trends.

When N is very large, A-SMVA can be efficient, because it avoids the iteration over n = 1 → N. However, we have two concerns about A-SMVA:

• The initial values of $Q_m^e(n)$ may significantly affect the outputs. For example, if they are too close to zero, the whole iteration will be skipped.

• The chosen value of 𝜖 may have a significant effect on the outputs. If 𝜖 is too big, we have less iterations, but sacrifice accuracy. If 𝜖 is too small, it may not converge in some instances (although we have not observed this).

As a summary, we provide one more numerically stable approach to determine the performance metrics for closed queueing networks with load-dependent queues. One can adopt SMVA or A-SMVA depending on the requirements.