An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue

Abstract

In many modern applications of queueing theory, the classical assumption of exponentially decaying service distributions does not apply. In particular, Internet and insurance risk problems may involve heavy-tailed distributions. A difficulty with heavy-tailed distributions is that they may not have closed-form, analytic Laplace transforms. This makes numerical methods, which use the Laplace transform, challenging. In this paper, we develop a method for approximating Laplace transforms. Using the approximation, we give algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy. We give several numerical examples, and we validate the approximation with known results where possible or with simulations otherwise. We also give convergence proofs for the methods.

A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

“…Since none of the moments exists in this case of Pareto inter-batch arrival time distribution, we calculate median from A(x) which is equal to 0.5 to consider an average of inter-batch arrival time as 20.882547 so that λ = 0.047887 and ρ = λḡ/μ * = 0.115012. We use geometric transform approximation method as described by Shortle et al (2004). As discussed by Shortle et al, we consider N = 100 probabilities y i = 1 − q i (1 ≤ i ≤ 100, 0 < q < 1).…”

Bayesian Analysis of a Queueing System with a Long-Tailed Arrival Process

“…Shortle et al (2004) propose a generalization of the TAM method called geometric TAM or GTAM. Their idea is to pick quantiles further out in the tail of the distribution.…”

“…If q is very close to 1 (0.999) the tail is not captured, but if it is close to the 0, the body of the distribution is not 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 F o r P e e r R e v i e w O n l y taken into account. In Shortle et al (2004), the authors propose choosing q so that the T AM mean matches the mean of the original distribution. Our experiments showed that in that way, the obtained value for r 0 (0.5872) is similar to that found with the original TAM but given the extra effort needed to find the optimal q, the computational time increases.…”

Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

“…Finally, I take up one of the numerical example used recently to demonstrate a new method for computing waiting-time distributions in M/G/1 queues with heavy-tailed service time distributions [20]. In the example considered here, the service times X i follow a unit Pareto distribution of the form…”

“…Depending on the problem, the improvement in computation time over the already highly efficient methods of Ref. [20] was seven to 58 fold (Tab. 3.4).…”