Erlang delay model: notation and derivation

Kendall–Lee notation

Erlang's delay model is

$$(M/M/N):(FCFS/\infty/\infty)$$

Poisson arrivals (rate $\lambda$), exponential holding time (mean $h = 1/\mu$), N servers, first-come-first-served, unlimited waiting room and infinite sources. Offered traffic $A = \lambda/\mu = \lambda h$ Erlangs, with $A < N$ for stability.

State diagram

State k = number of calls in the system (in service + waiting).

   lam     lam           lam      lam      lam
 0 ---> 1 ---> 2 ... N-1 ---> N ---> N+1 ---> ...
   <---   <---           <---     <---     <---
    mu    2mu            N mu     N mu     N mu

Arrival rate is always $\lambda$. Service rate is $k\mu$ for $k \le N$ and $N\mu$ for $k > N$ (only N servers).

Derivation

Balance between neighbouring states: $\lambda P_{k-1} = k\mu P_k$ for $k \le N$, and $\lambda P_{k-1} = N\mu P_k$ for $k > N$.

$$\begin{aligned} P_k &= \frac{A^k}{k!}P_0, & k \le N \\ P_k &= \frac{A^N}{N!}\left(\frac{A}{N}\right)^{k-N}P_0, & k > N \end{aligned}$$

Normalisation $\sum P_k = 1$ (the geometric series converges since $A/N < 1$):

$$\begin{aligned} P_0^{-1} &= \sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\sum_{j=0}^{\infty}\left(\frac{A}{N}\right)^j \\ &= \sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\,\frac{N}{N-A} \end{aligned}$$

A call is delayed ("blocked") when it finds all N servers busy, i.e. $k \ge N$:

$$\begin{aligned} P(W>0) &= \sum_{k=N}^{\infty}P_k = \frac{A^N}{N!}\,\frac{N}{N-A}\,P_0 \end{aligned}$$

Erlang C (delay) formula:

$$E_{2,N}(A) = \frac{\dfrac{A^N}{N!}\dfrac{N}{N-A}}{\displaystyle\sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\frac{N}{N-A}}$$

It can also be written using Erlang B: $E_{2,N} = \dfrac{N\,E_{1,N}}{N - A(1 - E_{1,N})}$.

Related results

• Mean delay of all calls: $W = E_{2,N}(A)\,\dfrac{h}{N-A}$
• Mean delay of delayed calls: $\dfrac{h}{N-A}$
• $P(\text{wait} > t) = E_{2,N}(A)\,e^{-(N-A)t/h}$
• For N = 1: $E_{2,1} = A$, $W = \dfrac{A\,h}{1-A}$ (M/M/1).