Pure loss system and Engset model

Pure loss system

A pure loss system is a system with $n$ servers and no queue. A call arriving when all $n$ servers are busy is lost (cleared) immediately, and the user does not retry. Its quality is measured by the probability of blocking (grade of service). Examples: a trunk group between two exchanges, a PABX with a few outgoing lines, the radio channels of a mobile cell. Erlang B (infinite sources) and Engset (finite sources) are the standard pure loss models.

Two blocking measures are used:

• Time congestion E: fraction of time all servers are busy.
• Call congestion B: fraction of call attempts that are lost.

Engset model

The Engset model is a pure loss model with a finite number of sources $N$ greater than the number of servers $n$ ($N > n$). It is used when $N$ is small, e.g. a PABX of 20 extensions sharing 5 trunks.

Assumptions

1. $N$ identical, independent sources.
2. An idle source makes calls at rate $\gamma$; a busy source makes no new call. So in state $k$ the arrival rate is $(N-k)\gamma$, which falls as $k$ increases.
3. Holding times are exponential with mean $h = 1/\mu$.
4. Blocked calls are cleared; full availability.

State diagram

   N.g      (N-1)g            (N-n+1)g
 0 -----> 1 -----> 2 ... n-1 ---------> n
   <-----   <-----             <-------
    mu       2mu                 n.mu

State probabilities. The cut equations give

$$(N-k)\gamma\,P(k) = (k+1)\mu\,P(k+1)$$

With $\beta = \gamma/\mu$ (offered traffic per idle source):

$$P(k) = \frac{\binom{N}{k}\beta^k}{\sum_{i=0}^{n}\binom{N}{i}\beta^i}, \qquad 0 \le k \le n$$

This is a truncated binomial distribution.

Time congestion

$$E = P(n) = \frac{\binom{N}{n}\beta^n}{\sum_{i=0}^{n}\binom{N}{i}\beta^i}$$

Call congestion. An arriving call comes from one of the idle sources, so it sees the system as if there were only $N-1$ sources:

$$B = E_n(N-1,\beta) = \frac{\binom{N-1}{n}\beta^n}{\sum_{i=0}^{n}\binom{N-1}{i}\beta^i}$$

Traffic. Carried traffic $Y = \sum k P(k)$; offered traffic per source $a = \beta/(1+\beta(1-B))$; offered traffic $A = N a$, and $A(1-B) = Y$.

Properties

• $B < E$ for finite sources, because when the group is busy fewer idle sources remain to call.
• As $N \to \infty$ with $N\gamma$ fixed, Engset tends to Erlang B.
• If $N \le n$ there is no blocking, and the distribution becomes the full binomial.
• Erlang B applied to a small group gives a pessimistic (too high) blocking, so Engset allows fewer trunks for the same GoS.