M/M/1 queue

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M/M/1 queue diagram
An M/M/1 queueing node

In queueing theory, a discipline within the mathematical theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times have an exponential distribution. The model name is written in Kendall's notation. The model is the most elementary of queueing models[1] and an attractive object of study as closed-form expressions can be obtained for many metrics of interest in this model. An extension of this model with more than one server is the M/M/c queue.

Model definition[edit]

An M/M/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.

  • Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
  • Service times have an exponential distribution with parameter 1/μ in the M/M/1 queue, where μ is the mean service rate.
  • A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
  • The buffer is of infinite size, so there is no limit on the number of customers it can contain.

The model can be described as a continuous time Markov chain with transition rate matrix

Q=\begin{pmatrix}
-\lambda & \lambda \\
\mu & -(\mu+\lambda) & \lambda \\
&\mu & -(\mu+\lambda) & \lambda \\
&&\mu & -(\mu+\lambda) & \lambda &\\
&&&&\ddots
\end{pmatrix}

on the state space {0,1,2,3,...}. This is the same continuous time Markov chain as in a birth–death process. The state space diagram for this chain is as below.

MM1 queue state space.svg

Transient solution[edit]

We can write a probability mass function dependent on t to describe the probability that the M/M/1 queue is in a particular state at a given time. We assume that the queue is initially in state i and write pk(t) for the probability of being in state k at time t. Then[2]

p_k(t)=e^{-(\lambda+\mu)t} \left[ \rho^{\frac{k-i}{2}} I_{k-i}(at) + \rho^{\frac{k-i-1}{2}} I_{k+i+1}(at) + (1-\rho) \rho^{k} \sum_{j=k+i+2}^{\infty} \rho^{-j/2}I_j(at) \right]

where \rho=\lambda/\mu, a=2\sqrt{\lambda\mu} and Ik is the modified Bessel function of the first kind. Moments for the transient solution can be expressed as the sum of two monotone functions.[3]

Stationary analysis[edit]

The model is considered stable only if λ < μ. If, on average, arrivals happen faster than service completions the queue will grow indefinitely long and the system will not have a stationary distribution. The stationary distribution is the limiting distribution for large values of t.

Various performance measures can be computed explicitly for the M/M/1 queue. We write ρ = λ/μ for the utilization of the buffer and require ρ < 1 for the queue to be stable. ρ represents the average proportion of time which the server is occupied.

Number of customers in the system[edit]

The probability that the stationary process is in state i (contains i customers, including those in service) is[4]:172–173

\pi_i=(1-\rho)\rho^i.\,

We see that the number of customers in the system is geometrically distributed with parameter 1 − ρ. Thus the average number of customers in the system is ρ/(1 − ρ) and the variance of number of customers in the system is ρ/(1 − ρ)2. This result holds for any work conserving service regime, such as processor sharing.[5]

Busy period of server[edit]

The busy period is the time period measured between the instant a customer arrives to an empty system until the instant a customer departs leaving behind an empty system. The busy period has probability density function[6][7][8][9]

f(t)=\begin{cases}
\frac{1}{t\sqrt{\rho}}e^{-(\lambda+\mu)t}I_1(2t\sqrt{\lambda\mu}) & t>0\\
0 & \text{otherwise}\end{cases}

where I1 is a modified Bessel function of the first kind,[10] obtained by using Laplace transforms and inverting the solution.[11]

The Laplace transform of the M/M/1 busy period is given by[12][13][14]:215

\mathbb E( e^{-\theta F} )= \frac{1}{2 \lambda}(\lambda + \mu + \theta - \sqrt{(\lambda + \mu + \theta)^2 - 4 \lambda \mu})

which gives the moments of the busy period, in particular the mean is 1/(μ − λ) and variance is given by

\frac{1+\frac{\lambda}{\mu}}{\mu^2(1-\frac{\lambda}{\mu})^3}.

Response time[edit]

The average response time or sojourn time (total time a customer spends in the system) does not depend on scheduling discipline and can be computed using Little's law as 1/(μ − λ). The average time spent waiting is 1/(μ − λ) − 1/μ = ρ/(μ − λ). The distribution of response times experienced does depend on scheduling discipline.

First-come, first-served discipline[edit]

For customers who arrive and find the queue as a stationary process, the response time they experience (the sum of both waiting time and service time) has transform (μ − λ)/(s + μ − λ)[15] and therefore probability density function[16]

f(t)=
\begin{cases}
(\mu-\lambda)e^{-(\mu-\lambda)t} & t>0\\
0 & \text{otherwise.}
\end{cases}

Processor sharing discipline[edit]

In an M/M/1-PS queue there is no waiting line and all jobs receive an equal proportion of the service capacity.[17] Suppose the single server serves at rate 16 and there are 4 jobs in the system, each job will experience service at rate 4. The rate at which jobs receive service changes each time a job arrives at or departs from the system.[17]

For customers who arrive to find the queue as a stationary process, the Laplace transform of the distribution of response times experienced by customers was published in 1970,[17] for which an integral representation is known.[18] The waiting time distribution (response time less service time) for a customer requiring x amount of service has transform[4]:356

W^\ast(s|x) = \frac{(1-\rho)(1-\rho r^2)e^{-[\lambda(1-r)+s]x}}{(1-\rho r^2)-\rho(1-r)^2e^{-(\mu/r-\lambda r)x}}

where r is the smaller root of the equation

\lambda r^2 - (\lambda + \mu + s)r + \mu = 0.

The mean response time for a job arriving and requiring amount x of service can therefore be computed as x μ/(μ − λ). An alternative approach computes the same results using a spectral expansion method.[5]

Diffusion approximation[edit]

When the utilisation ρ is close to one the process can be approximated by a reflected Brownian motion with drift parameter λ – μ and variance parameter λ + μ. This heavy traffic limit was first introduced by John Kingman.[19]

References[edit]

  1. ^ Sturgul, John R. (2000). Mine design: examples using simulation. SME. p. vi. ISBN 0-87335-181-9. 
  2. ^ Kleinrock, Leonard (1975). Queueing Systems Volume 1: Theory. p. 77. ISBN 0471491101. 
  3. ^ Abate, J.; Whitt, W. (1987). "Transient behavior of the M/M/l queue: Starting at the origin". Queueing Systems 2: 41. doi:10.1007/BF01182933.  edit
  4. ^ a b Harrison, Peter; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison–Wesley. 
  5. ^ a b Guillemin, F.; Boyer, J. (2001). "Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory". Queueing Systems 39 (4): 377. doi:10.1023/A:1013913827667.  edit
  6. ^ Abate, J.; Whitt, W. (1988). "Simple spectral representations for the M/M/1 queue". Queueing Systems 3 (4): 321. doi:10.1007/BF01157854.  edit
  7. ^ Keilson, J.; Kooharian, A. (1960). "On Time Dependent Queuing Processes". The Annals of Mathematical Statistics 31 (1): 104–112. JSTOR 2237497.  edit
  8. ^ Karlin, Samuel; McGregor, James (1958). "Many server queueing processes with Poisson input and exponential service times". Pacific J. Math. 8 (1): 87–118. doi:10.2140/pjm.1958.8.87. MR 0097132. 
  9. ^ Gross, Donald; Shortle, John F.; Thompson, James M.; Harris, Carl M. "2.12 Busy-Period Analysis". Fundamentals of Queueing Theory. Wiley. ISBN 1118211642. 
  10. ^ Adan, Ivo. "Course QUE: Queueing Theory, Fall 2003: The M/M/1 system". Retrieved 2012-08-06. 
  11. ^ Stewart, William J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press. p. 530. ISBN 0-691-14062-6. 
  12. ^ Asmussen, S. R. (2003). "Queueing Theory at the Markovian Level". "Applied Probability and Queues". Stochastic Modelling and Applied Probability 51. pp. 60–31. doi:10.1007/0-387-21525-5_3. ISBN 978-0-387-00211-8.  edit
  13. ^ Adan, I.; Resing, J. (1996). "Simple analysis of a fluid queue driven by an M/M/1 queue". Queueing Systems 22: 171. doi:10.1007/BF01159399.  edit
  14. ^ Kleinrock, Leonard (1975). Queueing Systems: Theory, Volume 1. Wiley. ISBN 0471491101. 
  15. ^ Harrison, P. G. (1993). "Response time distributions in queueing network models". "Performance Evaluation of Computer and Communication Systems". Lecture Notes in Computer Science 729. pp. 147–164. doi:10.1007/BFb0013852. ISBN 3-540-57297-X.  edit
  16. ^ Stewart, William J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press. p. 409. ISBN 0-691-14062-6. 
  17. ^ a b c Coffman, E. G.; Muntz, R. R.; Trotter, H. (1970). "Waiting Time Distributions for Processor-Sharing Systems". Journal of the ACM 17: 123. doi:10.1145/321556.321568.  edit
  18. ^ Morrison, J. A. (1985). "Response-Time Distribution for a Processor-Sharing System". SIAM Journal on Applied Mathematics 45 (1): 152–167. doi:10.1137/0145007. JSTOR 2101088.  edit
  19. ^ Kingman, J. F. C.; Atiyah (October 1961). "The single server queue in heavy traffic". Mathematical Proceedings of the Cambridge Philosophical Society 57 (4): 902. doi:10.1017/S0305004100036094. JSTOR 2984229.  edit