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=== Transition Matrix ===
The transition probability matrix for a M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below <ref name=":1" />:
<math>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}</math> ， <math>a_n=\frac{\lambda^n}{n!}e^{-\lambda}</math>, n = 0,1,....
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==Stationary distribution==
The number of jobs in the queue can be written as [[M/G/1 queue|M/G/1 type Markov chain]] and the stationary distribution found for state ''i'' (written π<sub>''i''</sub>) in the case ''D'' = 1 to be<ref name=":1">{{cite journal| url = http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.48_2_111.pdf | journal = Journal of the Operations Research Society of Japan | volume = 48 | year = 2005 | issue = 2 | pages = 111–122 | title = On the Series Expansion for the Stationary Probabilities of an M/D/1 queue | first = Kenji | last = Nakagawa}}</ref>
:<math>\begin{align}\pi_0 &= 1-\lambda \\
\pi_1 &= (1-\lambda)(e^\lambda - 1)\\
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