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Improving routing decisions in parallel non-observable queues

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Abstract

We revisit the well-known problem of scheduling in \(N\ge 2\) non-observable parallel single-server queues with one dispatcher having no queue to store the incoming jobs. The dispatcher does not observe the current states of the queues and servers and the sizes of the incoming jobs. The only available information to it is the job size distribution, job’s inter-arrival time distribution and server’s speeds. For this problem setting it is known that if the dispatcher can memorize the sequence of its previous decisions, then a deterministic policy is better than the random choice policy with respect to the job’s mean waiting and mean sojourn time. In this paper we address the following question: can the deterministic policy be improved if the dispatcher, in addition to the decisions made, can memorize also the times between the decisions? We describe the new policy and numerically show that it is almost always possible to do better with it than with the deterministic policy. Two algorithms for choosing decisions under the new policy are given. Both are based on the Lindley recursion and are approximate in nature, because utilize the discretization of the underlying distributions. Our findings show that the relative gain of the new policy may reach \(10\%\), when minimizing job’s mean sojourn time, and more than \(50\%\) for the job’s mean waiting time.

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Notes

  1. It means that the dispatcher does not have a queue to store the jobs.

  2. Even so, discrete probability distributions are also allowed. The numerical procedures will require some refinements in order to align the discrete grids.

  3. Because of the constraints, which are put by the assumptions on the admissible policies, sometimes in the literature the latter are called “blind” scheduling policies.

  4. Discretization is also justified by the fact that in practice, there exists a minimal time gap, when nothing happens in the system: minimal inter-arrival time or minimal service time. This minimal value can be taken as the discretization step size. As it will be seen from the experimental results, we may even a larger value for the step size without any harm for the results.

  5. In (4)–(7) and further, where necessary, the sign “\(\approx \)” is used instead of “\(=\)” since the inter-arrival times \(\tau _n\) may not be the multiple of \(\varDelta \). That is why the rounding operation is applied.

  6. The following observation is worth noticing here. For the two single-server fully observable queues, operating in parallel, it is known that the optimal policy, with respect to the mean sojourn time, is of threshold type (see a short review in [26] and references therein). For the general case, when the number of queues is greater than two, the structure of the optimal policy is not known. Here the constant \(\theta \) can be considered, to some extent, as a threshold, which introduction leads to the policy improvement.

  7. The literature in this topic is very rich. We can refer, for example, to [11, 32, 35].

  8. The value of \(\varDelta \) must be less than than both the expected inter-arrival time and the expected time between service completions across all the servers.

  9. For example, for the two-server system the optimal deterministic policy always exists. In the case \(N\rightarrow \infty \), it is more reasonable to develop approximations instead of using exact formulae (see [5]).

  10. Some DET policies may be periodic, others may not. For example, for the two single-server queues in parallel with exponential service at rates 1 and 2 respectively and Poisson flow at rate 1, the optimal DET policy with respect to the mean sojourn time is the periodic policy \((11011011101101101110)^\infty \). Here 0 means that the job is routed to the server with rate 1. We can imagine a corresponding RND policy not with one parameter \(p_0\) (\(p_1=1-p_0\)), but with \(2^{19}\) parameters. It remains an open question whether such an RND policy can do as good as DET or, in general, whether it can be better than DET, when the latter is not optimal.

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (Grant 18-07-00692).

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Authors and Affiliations

  1. Institute of Informatics Problems of the FRC CSC RAS, 44k2 Vavilova Street, Moscow, Russian Federation, 119333

    Mikhail Konovalov & Rostislav Razumchik

  2. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198

    Rostislav Razumchik

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  1. Mikhail Konovalov
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Konovalov, M., Razumchik, R. Improving routing decisions in parallel non-observable queues. Computing 100, 1059–1079 (2018). https://doi.org/10.1007/s00607-018-0598-5

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