Response times in M/M/s fork-join networks

Abstract

We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

Exact Monte Carlo simulation for fork-join networks

“…In Table 1 we present the Monte Carlo estimates of E W for the fork-join network with K = 2, λ = 1, s i = 1, and different values of µ i and N . From Table 1 we can see that the approximations, given in [9] with N = ∞, are very close to the Monte Carlo estimates using perfect simulation with a finite value of N. Obviously, the fork-join network with N = ∞, on average, has a longer queue at equilibrium than the fork-join network with a finite value of N . Carlo estimate with N = 300 and the dashed line is the approximation in [9] with N = ∞.…”

“…We can also compare the Monte Carlo estimates and the approximation in [9] for F (t) = P(W ≤ t). With µ 1 = 10 and µ 2 = 20, the results are shown in Figure 2, where the solid line (the Monte Carlo estimate) and the dashed line (the approximation given in [9]) are almost the same. If we zoom into the graph (see Figure 3), we can see that the approximation is slightly lower.…”

“…Based on the simulated realisations we provide Monte Carlo estimates for the distributions of the response time and queue length. Comparisons of the Monte Carlo estimates based on perfect sampling and approximation results in [9] demonstrate that the proposed perfect sampling method works very well.…”

“…From the results in [9] we know that when N = ∞, the transition rate for the K-dimensional continuous-time Markov chain Q(t) from a state n = (n 1 , . .…”

A Maclaurin-series expansion approach to multiple paired queues

“…Finally, we note that the paired queueing system somewhat resembles a fork-join queueing system; see e.g. [6] and the references therein. However, in fork-join queueing systems both arrivals and departures in the different buffers are synchronised, which leads to entirely different dynamics.…”