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. 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.

Queue size at arrival times

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

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

where $a_{v}=\mathbb {E} \left({\frac {(\mu X)^{v}e^{-\mu A}}{v!}}\right)$.

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

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

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

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.

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.