
TL;DR
This paper investigates the distribution of maximum queue length in a random customer-server model, highlighting the complexity of extending known results from simpler systems and discussing potential generalizations.
Contribution
It provides algebraic expressions for maximum queue length in a two-server system and explores the challenges of extending these to three servers, reviewing related queue wait time distributions.
Findings
Algebraic expressions derived for two-server maximum queue length.
Complexity increases significantly with three servers, making algebraic solutions difficult.
Discussion on generalizing maximum wait time distributions from M/M/1 queues.
Abstract
In discrete time, customers arrive at random. Each waits until one of three servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. Algebraic expressions obtained for the two-server scenario do not appear feasible here. We also review well-known distributional results for maximum wait time associated with an M/M/1 queue and speculate about their generalization.
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Taxonomy
TopicsAdvanced Queuing Theory Analysis · Probability and Risk Models · Simulation Techniques and Applications
How Long Might We Wait at Random?
Steven Finch
(August 27, 2022)
Abstract
In discrete time, customers arrive at random. Each waits until one of three servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. Algebraic expressions obtained for the two-server scenario do not appear feasible here. We also review well-known distributional results for maximum wait time associated with an M/M/ queue and speculate about their generalization.
00footnotetext: Copyright © 2019 & 2022 by Steven R. Finch. All rights reserved.
In queueing theory, maximum line length and maximum wait time are different sides of the same coin, one spatial (Section 1) and the other temporal (Section 2).
1 In Spatium
Let and . Consider the Julia program:
[TABLE]
which simulates the maximum value of a Geo/Geo/ queue with LAS-DA over time steps. The Boolean expressions containing Julia’s Uniform random deviate generator ensure that Bernoulli() and Bernoulli(). The word “Geometric” arises because
[TABLE]
where and
[TABLE]
where . Clearly . LAS stands for “late arrival system” and DA stands for “delayed access” [1]; in particular, a customer entering an empty queue at time is not immediately eligible for service, but rather at time . We study the asymptotic distribution of the maximum . This section is best thought of as an extended addendum to [2], written for the sake of completeness. The only difference with our earlier work is that closed-form expressions here become unwieldy and thus our approach is more numeric and less symbolic.
The Poisson clumping heuristic [3], while not a theorem, gives results identical to exact asymptotic expressions when such exist, and evidently provides excellent predictions otherwise. Consider an irreducible positive recurrent Markov chain with stationary distribution . For large enough , the maximum of the chain satisfies
[TABLE]
as , where is the sojourn time in during a clump of nearby visits to .
Starting with transition matrix
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we obtain [4]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
and satisfies the cubic equation , that is,
[TABLE]
where
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Note that, if , we have
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thus
[TABLE]
We need now to calculate . Consider a random walk on the integers consisting of incremental steps satisfying
[TABLE]
For nonzero , let denote the probability that, starting from , the walker eventually hits [math]. Let denote the probability that, starting from [math], the walker eventually returns to [math] (at some future time). We have two values for : when it is used in a recursion, it is equal to ; when it corresponds to a return probability, it retains the symbol . Let . Using
[TABLE]
[TABLE]
define
[TABLE]
equivalently
[TABLE]
equivalently
[TABLE]
equivalently
[TABLE]
Examine the denominator of . Only the first three of its four zeroes , , , are of interest (the fourth is ). Substituting , and into the numerator of , then setting , gives three equations in four unknowns.
Let . Using
[TABLE]
[TABLE]
define
[TABLE]
equivalently
[TABLE]
equivalently
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equivalently
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Examine the denominator of . Only the zero of smallest modulus interests us. Substituting into the numerator of , and setting , gives a fourth equation (to include with the other three from earlier). For example, if
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we have
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from earlier and
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after solving the simultaneous system in , , , . Observe that
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as . The ratio within the exponential argument is
[TABLE]
and hence
[TABLE]
for sufficiently large , where denotes Euler’s constant [5]. Such moment formulas usually contain tiny periodic fluctuations, but we omit these from consideration.
The use of an expected maximum for performance analysis, instead of a simple average, does not appear to lead to surprising outcomes. A corollary of the preceding numerical results is that, in a busy hospital emergency room (with ), one fast doctor (with ) outperforms three slow doctors (each with ). For average queue lengths [4],
[TABLE]
corresponding to Geo/Geo/ and
[TABLE]
corresponding to Geo/Geo/. This is also consistent with results in [6] governing deterministic traffic signals: we do better with an pattern than with .
2 In Tempore
Let be an integer and . Consider the R program:
[TABLE]
which simulates the maximum wait time associated with an M/M/ queue over the time interval . More precisely, at any arrival time , let denote the patient wait time in the system (either queue or treatment) and denote the patient wait time in the queue (excluding treatment). For , the maximums of and over all arrival times up to satisfy [7, 8, 9]
[TABLE]
[TABLE]
where . These imply
[TABLE]
[TABLE]
for large enough . Also, if and , then
[TABLE]
[TABLE]
In particular, assuming , the expected maximum wait times are and respectively.
For , no analogous formulas are known. Assuming , and , we estimate via simulation that
[TABLE]
for and
[TABLE]
for . With regard to the expected maximum of , one fast doctor outperforms slow doctors. With regard to the expected maximum of , however, slow doctors outperform one fast doctor. Experimental histograms of such maximums appear to be roughly Gumbel-shaped: a two-moment fit provides a fairly compelling but ultimately inconclusive match between theory and data. Only rigorous analysis for will clarify the situation.
The same trend occurs for simple averages. Given , as before and allowing , we have the following when :
[TABLE]
when :
[TABLE]
and when :
[TABLE]
Such mean results apply not only to the FIFO discipline (first in, first out), but also to the LIFO (last in, first out) and SIRO (serve in random order) disciplines. The underlying SIRO wait time distribution is more complicated than that for FIFO, as shown for in [10, 11].
Alternative treatments of M/M/ maximum wait times include [12, 13], where or refer not to the length of the time interval but instead the total number of customers. Simple averages of wait times for Geo/Geo/ can be calculated by methods given in [14]; maximums remain open.
3 Acknowledgements
I am thankful to Guy Louchard for introducing me to the Poisson clumping heuristic (especially recursions for and ), and to Stephan Wagner for extracting discrete Gumbel asymptotics in [15] (a contribution leading to [2, 16, 17]). See also [18, 19] for work involving the case of Uniform service times and Deterministic subcase (). The creators of R, Julia, Mathematica and Matlab, as well as administrators of the MIT Engaging Cluster and the MIT Supercloud Cluster, earn my gratitude every day.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] J. J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2: Discrete Time Models: Techniques and Applications , Academic Press, 1983, pp. 189–200; MR 0719019.
- 2[2] S. Finch, Geo/Geo/ 2 2 2 queues and the Poisson clumping heuristic, ar Xiv:1902.09272.
- 3[3] D. Aldous, Probability Approximations via the Poisson Clumping Heuristic , Springer-Verlag, 1989, pp. 1–8, 23–25, 30; MR 0969362.
- 4[4] J. R. Artalejo and O. Hernández-Lerma, Performance analysis and optimal control of the Geo/Geo/ c 𝑐 c queue, Performance Evaluation 52 (2003) 15–39.
- 5[5] S. R. Finch, Euler-Mascheroni constant, Mathematical Constants , Cambridge Univ. Press, 2003, pp. 28–40; MR 2003519.
- 6[6] S. Finch and G. Louchard, Conjectures about traffic light queues, ar Xiv:1810.03906.
- 7[7] D. L. Iglehart, Extreme values in the GI/G/ 1 1 1 queue, Annals Math. Statist. 43 (1972) 627–635; MR 0305498
- 8[8] D. L. Iglehart, Regenerative Simulation for Extreme Values , Stanford Univ. Technical Report 43 (1977); http://apps.dtic.mil/docs/citations/ADA 047945.
