G/M/1 queue

In queueing theory, a discipline within the mathematical theory of probability, the G/M/1 queue represents the queue length in a system where interarrival times have a general (meaning arbitrary) distribution and service times for each job have an exponential distribution.[1] The system is described in Kendall's notation where the G denotes a general distribution, M the exponential distribution for service times and the 1 that the model has a single server.

The arrivals of a G/M/1 queue are given by a renewal process. It is an extension of an M/M/1 queue, where this renewal process must specifically be a Poisson process (so that interarrival times have exponential distribution).

Models of this type can be solved by considering one of two M/G/1 queue dual systems, one proposed by Ramaswami and one by Bright.[2]

Queue size at arrival times

Let ${\displaystyle (X_{t},t\geq 0)}$ be a ${\displaystyle G/M(\mu )/1}$ queue with arrival times ${\displaystyle (A_{n},n\in \mathbb {N} )}$ that have interarrival distribution A. Define the size of the queue immediately before the nth arrival by the process ${\displaystyle U_{n}=X_{A_{n}-}}$. This is a discrete-time Markov chain with stochastic matrix:

${\displaystyle P={\begin{pmatrix}1-a_{0}&a_{0}&0&0&0&\cdots \\1-(a_{0}+a_{1})&a_{1}&a_{0}&0&0&\cdots \\1-(a_{0}+a_{1}+a_{2})&a_{2}&a_{1}&a_{0}&0&\cdots \\1-(a_{0}+a_{1}+a_{2}+a_{3})&a_{3}&a_{2}&a_{1}&a_{0}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{pmatrix}}}$

where ${\displaystyle a_{v}=\mathbb {E} ({\frac {(\mu X)^{v}e^{-\mu A}}{v!}})}$.[3]^:427–428

The Markov chain ${\displaystyle U_{n}}$ has a stationary distribution if and only if the traffic intensity ${\displaystyle \rho =(\mu \mathbb {E} (A))^{-1}}$ is less than 1, in which case the unique such distribution is the geometric distribution with probability ${\displaystyle \eta }$ of failure, where ${\displaystyle \eta }$ is the smallest root of the equation ${\displaystyle \mathbb {E} (\exp(\mu (\eta -1)A))}$.[3]^:428

In this case, under the assumption that the queue is first-in first-out (FIFO), a customer's waiting time W is distributed by:[3]^:430

${\displaystyle \mathbb {P} (W\leq x)=1-\eta \exp(-\mu (1-\eta )x)~{\text{ for }}x\geq 0}$

Busy period

The busy period can be computed by using a duality between the G/M/1 model and M/G/1 queue generated by the Christmas tree transformation.[4]

Response time

The response time is the amount of time a job spends in the system from the instant of arrival to the time they leave the system. A consistent and asymptotically normal estimator for the mean response time, can be computed as the fixed point of an empirical Laplace transform.[5]

References

1. Adan, I.; Boxma, O.; Perry, D. (2005). "The G/M/1 queue revisited" (PDF). Mathematical Methods of Operations Research. 62 (3): 437. doi:10.1007/s00186-005-0032-6.
2. Taylor, P. G.; Van Houdt, B. (2010). "On the dual relationship between Markov chains of GI/M/1 and M/G/1 type" (PDF). Advances in Applied Probability. 42: 210. doi:10.1239/aap/1269611150.
3. Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes (second ed.). Oxford University Press. ISBN 0198572220.
4. Perry, D.; Stadje, W.; Zacks, S. (2000). "Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility". Operations Research Letters. 27 (4): 163. doi:10.1016/S0167-6377(00)00043-2.
5. Chu, Y. K.; Ke, J. C. (2007). "Interval estimation of mean response time for a G/M/1 queueing system: Empirical Laplace function approach". Mathematical Methods in the Applied Sciences. 30 (6): 707. doi:10.1002/mma.806.