M/D/1 queue

In queueing theory, a discipline within the mathematical theory of probability, an M/D/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 are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue.

Model definition

An M/D/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 are deterministic time D (serving at rate μ = 1/D).
• 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 state space diagram for M/D/1 queue is as below:

Stage Space Diagram of M/D/1 Queue

Transition Matrix

$P={\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&...\\a_{0}&a_{1}&a_{2}&a_{3}&...\\0&a_{0}&a_{1}&a_{2}&...\\0&0&a_{0}&a_{1}&...\\...&...&...&...&...\\\end{pmatrix}}$ ， $a_{n}={\frac {\lambda ^{n}}{n!}}e^{-\lambda }$, n = 0,1,....

Classic performance metrics

$L=\rho +{\frac {1}{2}}\left({\frac {\rho ^{2}}{1-\rho }}\right);$

$L_{Q}={\frac {1}{2}}\left({\frac {\rho ^{2}}{1-\rho }}\right);$

$\omega ={\frac {1}{\mu }}+{\frac {\rho }{2\mu (1-\rho )}};$

$\omega _{Q}={\frac {\rho }{2\mu (1-\rho )}}$

Example

Customers arrive a Starbucks line at a rate of 20 per hour, and follows an exponential distribution. There is only one server, the service rate is at a constant of 30 per hour.

Arrival Rate: 20 per hour

Service Rate: 30 per hour

ρ=20/30=2/3

Using the queueing theory equations, the results are as following:

Average number in line= 0.6667

Average number in system: 1.333

Average time in line: 0.033 hour

Average time in system: 0.067 hour

Relation for Mean Waiting Time in M/M/1 and M/D/1 queues

For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below:

$E(W)={\frac {\rho \tau }{2(1-\rho )}}(1+{\frac {\sigma ^{2}}{\tau ^{2}}})$

where τ is the mean service time; ${\textstyle \sigma ^{2}}$ is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers.

For M/M/1 queue, the service times are exponentially distributed, then ${\textstyle \sigma ^{2}}$=${\textstyle \tau ^{2}}$ and the mean waiting time in the queue denoted by W_M is given by the following equation:

${W_{M}}={\frac {\rho \tau }{1-\rho }}$

Using this, the corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e.  ${\textstyle \sigma ^{2}}$=0.  The mean waiting time in the M/D/1 queue denoted as W_D is given by the following equation:

${W_{D}}={\frac {\rho \tau }{2(1-\rho )}}$

From the two equations above, we can infer that Mean queue length in M/M/1 queue is twice that in M/D/1 queue.

Stationary distribution

The number of jobs in the queue can be written as an M/G/1 type Markov chain and the stationary distribution found for state i (written π_i) in the case D = 1 to be

${\begin{aligned}\pi _{0}&=1-\lambda \\\pi _{1}&=(1-\lambda )(e^{\lambda }-1)\\\pi _{n}&=(1-\lambda )\left(e^{n\lambda }+\sum _{k=1}^{n-1}e^{k\lambda }(-1)^{n-k}\left[{\frac {(k\lambda )^{n-k}}{(n-k)!}}+{\frac {(k\lambda )^{n-k-1}}{(n-k-1)!}}\right]\right)\quad (n\geq 2).\end{aligned}}$

Delay

Define ρ = λ/μ as the utilization; then the mean delay in the system in an M/D/1 queue is

${\frac {1}{2\mu }}\cdot {\frac {2-\rho }{1-\rho }}.$

and in the queue:

${\frac {1}{2\mu }}\cdot {\frac {\rho }{1-\rho }}.$

Busy period

The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to D times the number of customers served. If ρ < 1, then the number of customers served during a busy period of the queue has a Borel distribution with parameter ρ.

Finite capacity

Stationary distribution

A stationary distribution for the number of customers in the queue and mean queue length can be computed using probability generating functions.

$P_{0}(N)={\frac {1}{1+\rho b_{N-1}}};$

$P_{N}(N)=1-{\frac {b_{N-1}}{1+\rho b_{N-1}}};$

$P_{j}(N)={\frac {b_{j}-b_{j-1}}{1+\rho b_{N-1}}}$， j = 1,..., N-1.

Transient solution

The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al in 2002.

The mean number of customers in M/D/1/N queue presented in Garcia et al. 2002 is as follows:

$X_{N}=N-{\frac {\sum _{k=0}^{N-1}b_{k}}{1+\rho b_{N-1}}};$

The mean waiting time W N in the M/D/1/N queuing system presented in Garcia et al. 2002 is as follows:

$W_{N}=(N-1-{\frac {\sum _{k=0}^{N-1}b_{k}-N}{\rho b_{N-1}}})T$

Application

Includes applications in wide area network design, where a single central processor to read the headers of the packets arriving in exponential fashion, then computes the next adapter to which each packet should go and dispatch the packets accordingly. Here the service time is the processing of the packet header and cyclic redundancy check, which are independent of the length of each arriving packets. Hence, it can be modeled as a M/D/1 queue.