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A Queueing System with Determined Delay in Starting the Service
Alexey S. Eremin Saint-Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia [email protected]
Abstract. A one-server queuing system with general distribution of service time is considered. An approximation to the general distribution with two-stage law the where first stage is deterministic and the second is exponential is suggested. It is shown that the coefficient of variation can be made arbitrarily small by choosing the parameters of the suggested distribution. The steady-state and transient equations are presented. The transient behavior is described with a system of delay-differential equations. Imitation results are compared to the obtained steady-state equations solution.
Keywords: queueing systems, coefficient of variation, two-stage service, deterministic time, transient system, delay differential equations.
Introduction
In the paper we consider queueing systems with exponentially distributed customers arrival times and a single server. General distribution of service time (G in M / G /1 Kendall’s notation) lead to non-Markovian processes and their transient behaviour cannot be described with ordinary differential equations (ODEs). There exists a quite complicated procedure for determining steady-state probabilities and even more complicated way to compute non-stationary probabilities for the case of discrete Markov chains (see eg. [1]).
In order to make the analysis and computations easier the service distribution is approximated usually with к-stage Erlang ( M/Ek /1), hyperexponential ( M/H2/I) or Coxian (M / C /1) distributions, in which the service consists of several stages with exponentially distributed times. Thus the overall process becomes Markovian and a system of ODEs describes its transient behaviour. However, it can be shown that the Erlang distribution with к exponential stages of rates ц, i = 1,...,к can only approximate the distribution with the coefficient of variation (CV) greater than 1/-\/k. Otherwise the rates ц must be taken negative or complex (which makes no physical sense, though is a smart mathematical instrument, if one is able to interpret obtained results). A similar situation with small CVs appears in other mentioned distributions [2].
The extreme case of CV equal to 0 means that the time is fixed and its probability density function (PDF) is 5(t), where T is the fixed service time and 5(x) is Dirac delta function. In this case the process is denoted as M / D /1, where D stands for “determined time”. Results for such models with finite capacity queue were obtained by Brun et al. for both the steady-state
[3] and the transient [4] behaviour. The latter is described with delay-differential equations (DDEs).
In order to approximate small CVs a so-called “hyper-delta” distribution was proposed in [5]. Its PDF is a linear combination
of several delta functions and the cumulative distribution function (CDF) looks like a “stair-case”. Indeed, the CV of such distribution can be made as small as needed and the physical sense of every probability is clear. Still there are two issues. First, the transient system is a multiple delay differential equation, so its numerical solution demands special methods [6] (while a single fixed delay equation can be solved with classical methods with adapted step-size control). Second, a discrete random variable (despite of the fact that it can provide any number of momenta to have demanded values) is significantly different from a continuous one in physical sense.
Considering all the mentioned we suggest to use a two-stage service with first stage proceeded in determined time T and the second in exponentially distributed time with rate ц. The CDF of both stages combined is
F(t) = u(t - T)(1 - е~ц(-T>), (1)
which now is continuous in contrast to the hyper-delta distribution, and PDF is f (t)= u (t - T)це ц^ T^. Since this is just exponential distribution shifted by T, its mean value is T +1/ц and the standard deviation is the same as for the exponential distribution, namely 1/ц . CV then has the form
1 + ЦГ
and can be made as small as needed.
It should be mentioned that mostly we are interested in the study of transient behaviour, from one hand since the steady-state case can be derived from general formulae with F (t) substituted from (1), and from the other hand since much attention has been recently devoted to the analysis of critical loads in queuing systems (eg. [7]), i. e. to the study of their non-stationary characteristics.
A SYSTEM WITHOUT A QUEUE
Let’s start with a simple example to get the general idea of using a determined time stage. Consider a service system with no waiting room at all. Customers arrive according to a Poisson process at rate X. The service is made in two stages: first, the same constant time T for each customeris spent, and the second stage time is distributed with another Poisson law with rate ц . Denote the state with no customers in the system as {}, when a customer is at the first stage of service as {1,1} and at the second stage as {1,2}. The probabilities of the states are p0, P11 and P12 respectively. The transition diagram is presented at Fig. 1.
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Fig. 1. M/R/1/0 system transition diagram
P0 (t)=-XP0 (t)+МР1,2 (t)
P\,1 (t) = Vo (t)+ PP2,2 (t)-Vo (t -T) -PP2,2 (t - T),
P1,2 (t) = e~XT *(XP0 (t - T)+ PP2,2 (t - T)) -XP1,2 (t)-PP1,2 (t)•
The transition from the state {1,1} to {1, 2} is shown with double arrow and time T is taken in square brackets to differ it from transition rates X and p. We suggest denoting the system as M / R /1/0 with R standing for “retarded”.
Assuming that the initial state of the system is P0 (0) = 1 and all other probabilities equal to 0 we can write down the system of Chapman-Kolmogorov equations. In this case it is a retarded delay differential equation system
P0 (t)=-XP0 (t)+m,2 (t)
< P1,1 (t) = XP0 (t)-XP0 (t -T )u (t -T ) (2)
_P1,2 (t)= XP0 (t -T)u (t -T)-PP1,2 (t).
However, if we assume that all variables are strictly 0 for any t < 0 (thus a jump discontinuity for p0 (t) in t = 0 appear), we can omit writing down u (t - T). In all following equations such assumption is made.
Properties of the system can be easily derived. The average service time is T +1/ p and the probability of the system being
, • T + 1/p
busy is —----------— .
1/X + T + 1/p
Unbounded queue
Now we consider an infinite capacity queueing system with the same two-stage service (M / R /1 -system in our denotations). The assumption X < p/(1 + pT) is made to regulate the size of the queue. The transition diagram is presented at Fig. 2. As soon as one customer leaves the queue a new one comes to the “delay” stage of service.
Two arrows from {k,1} to {k +1,1} and {k, 2} to {k +1, 2} states actually mean that in any moment from the start of serving k -th customer another one can come into the queue. (A wide arrow covering the whole square between the four states would probably show it better, but it would definitely make a mess of the diagram).
In any moment during the time T a customer is at the first stage of service a new customer may enter the queue.
Then the rate of transition from {1,1} state to {1, 2} is lower than in the system without a queue. Comparing to (2) the equation for the same three probabilities are
The writing can be simplified if we denote a “pure input function” into the state {1,1} as s! (t )= Xf>0 (t)+ pP2 2 (t).
It takes into account all transitions that lead the system to the very beginning of the {1,1} state when no time of the first service stage has yet passed. In the case of the state {1,1} there are no other ways into it but from {0} and {2, 2}.
However, for all other {k,1} states (k > 1) such “pure input” is Sk (t)= PPk+1 2 (t), while the transfer from {k -1,1} made during the time of first service stage is not included.
Full transient system
Instead of writing down the system for every Pk1 (t) and Pk 2 (t) as in (3), we can make it much simpler if we introduce the probability of k customers being in the system as Pk (t) = Pk 1 (t)+ Pk 2 (t) and writing down the system for Pk,2 (t) and Pk (t) ■
P0 (t)=-XP0 (t)+PP1,2 (tX
P1,2 (t)= e~XTs1 (t - T)-(X + P)P1,2 (tX
P2,2 (t) = e-XT (S2 (t - T)+ XTS1 (t - T))
-(X + P)P2,2 (t> (4)
. k-1
Pk (t) = e Z (XT) sk-i (t - T)-(X + p)P2,2 (t X
i=0
and
p>1 (t)= S1 (t)-XP1 (t)-PP12 (tX
P2 (t) = s2 (t)+XP1 (t)-XP2 (t)- PP2,2 (tX <... (5)
Pk (t)= sk (t)+XPk-1 (t)-XPk (t)- PPk,2 (t)
The values of Pkl (t) can be then found as Pk (t)-Pk 2 (t) if needed.
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The system (4) is independent from (5) and can be solved separately, leaving (5) to be just a system of ordinary differential equations.
Steady state probability
DISTRIBUTION
In the steady state p (t — T ) = p (t ) = p, and the algebraic system (4), (5) simplifies to
XP0 = dP1,2,
(X + d) P12 = e XP0 + e dP2,2,
(X + d)P2,2 = XTe~XTXP0 + e~XT (XTdP2,2 + dP3,2 X
* (X + d)Pk,2 = (XT)—1 e~XTXP0 (6)
+e~XT S(XT) dPk—+1,2,
i=0
XPk = dPk+1,2, k = 1 »
with an additional normalising condition ^”0 Pi = 1.
Using the formula for stationary probabilities of M / G /1 system [1] with substitution of the known F (t) we can compute
P0 as
P0 = 1 — XT — X/d . (7)
All other state probabilities Pi, i = 1,» are now much easier to compute from the system (6), than with application of the general formulae.
Imitation comparison
We have chosen the following parameters for a test computation of steady-state distribution: X = 2, T = 0.2, d = 4 (so, the coefficient of variation is 5 < -k i. e. smaller than the small-
9 v2
est value achievable with two-stage Erlang). Also the probabilities were calculated using an imitation model with 2 000 000 served customers. The results are presented in the Table 1. They are equal up to a certain accuracy.
Conclusion
Queueing models with deterministic times of customer arrivals or services or some stages of more complex service processes are described with systems of retarded delay differential equations. They are much simpler to solve numerically than systems corresponding to general distributions, which include integral-differential equations.
Comparison of the imitation to the system (6)
Number of customers in the system Probability by system (6) Probability by imitation
0 0.1 0.09942
1 0.12377 0.12319
2 0.11287 0.11264
3 0.09720 0.09735
4 0.08302 0.08304
5 0.07085 0.07103
6 0.06046 0.06048
7 0.05160 0.05165
8 0.04403 0.04410
9 0.03757 0.03777
10 0.03206 0.03217
A very simple example was considered in this paper. Still, the results can be extended to systems with several service channels with different delays, with non-constant delays depending on time (e. g. some periodical behaviour of enter or service probability distributions) or even on the state of the system itself (e. g. longer “warm-up” for “cold” systems). Such an approach allows better approximations to general distributions in case of small enough coefficients of variation.
References
1. Bocharov P. P., D’Apice C., Pechinkin A. V. Queueing Theory. Utrecht, Boston, Walter de Gruyte, 2004. 446 p.
2. Khomonenko A. D., Bubnov V P. A use of coxian distribution for iterative solution of M/G/N/R < да queuing systems. Probl. Of Control and Inform. Theory, 1985, Vol. 14, pp. 143-153.
3. Brun O., Garcia J.-M. Analytical solution of finite capacity M/D/1 queues. J. Appl. Prob., 2000, Vol. 37, pp. 1092-1098.
4. Garcia J.-M., Brun O., Gauchard D. Transient analytical solution of M/D/1/N queues. J. Appl. Prob., 2002, Vol. 39, pp. 853-864.
5. Smagin V. A. Gusenitsa Ya. N. On modelling of non-stationary one-server queueing system with arbitrary distributions of customers arrivals and services [O modelirovanii odnokanal’noi nestatsionarnoi sistemy s proizvol’nym raspredeleniem momen-tov vremeni postupleniya zayavok i ih obsluzhivaniya]. Inteltech. Technique of Communication [Intelteh. Tehnika Sredstv Svyazi], 2014, is. 3 (142), pp. 15-20.
6. Bellen A., Zennaro M. Numerical Methods for Delay Differential Equations. Oxford: Oxford Univ. Press, 2013. 395 p.
7. Bubnov V. P, Khomonenko A. D., Tyrva A. V. Software reliability model with coxian distribution of length of intervals between errors detection and fixing moments. Proceedings — 35 th Annual IEEE Int. Computer Software and Applications Conf. Workshops, COMPSACW, 2011, pp. 310-314.
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Система массового обслуживания с детерминированным временем задержки
начала обслуживания
Еремин А. С.
Санкт-Петербургский государственный университет 199034, Санкт-Петербург, Университетская наб., 7/9 [email protected]
Аннотация. Рассматривается одноканальная система массового обслуживания с общим законом распределения времени обслуживания. Общий закон предлагается приближать двухфазным распределением, первая фаза которого детерминированная, а вторая - экспоненциальная. Показано, что выбором параметров такого двухфазного распределения можно получить сколь угодно маленький коэффициент вариации. Представлены уравнения для переходного режима и для стационарного распределения вероятностей. Переходный процесс описывается системой дифференциальных уравнений с запаздывающим аргументом. Результаты имитационного моделирования сравниваются с решением системы на стационарные вероятности.
Ключевые слова: система массового обслуживания, коэффициент вариации, двухфазное обслуживание, детеминированное время, переходный режим, дифферецнальные уравнения с запаздывающим аргументом.
References
1. Bocharov P. P. Queueing Theory / P. P. Bocharov, C. D’Api-ce, A. V. Pechinkin. - Utrecht : Boston : Walter de Gruyte, 2004. -446 p.
2. Khomonenko A. D. A use of coxian distribution for iterative solution of M/G/N/R < да queuing systems / A. D. Khomonenko,
V P. Bubnov // Probl. Of Control and Inform. Theory. - 1985. -Vol. 14. - P. 143-153.
3. Brun O. Analytical solution of finite capacity M/D/1 queues / O. Brun, J.-M. Garcia // J. Appl. Prob. - 2000. - Vol. 37. -
P. 1092-1098.
4. Garcia J.-M. Transient analytical solution of M/D/1/N queues / J.-M. Garcia, O. Brun, D. Gauchard // J. Appl. Prob. -2002. - Vol. 39. - P. 853-864.
5. Смагин В. А. О моделировании одноканальной нестационарной системы с произвольным распределением моментов времени поступления заявок и их обслуживания / В. А. Смагин, Я. Н. Гусеница // Интелтех. Техника средств связи. - 2014. - Вып. 3 (142). - С. 15-20.
6. Bellen A. Numerical Methods for Delay Differential Equations / A. Bellen, M. Zennaro. - Oxford : Oxford Univ. Press,
2013. - 395 p.
7. Bubnov V. P. Software reliability model with coxian distribution of length of intervals between errors detection and fixing moments / V. P. Bubnov, A. D. Khomonenko, A. V. Tyr-va // Proceedings - 35th Annual IEEE Int. Computer Software and Applications Conf. Workshops. - COMPSACW. - 2011. -Р. 310-314.
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