 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-37/1/i/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 37 / 1 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

A loss network models calls that need several resources simultaneously and are rejected, rather than queued, when insufficient capacity remains. Take finite resource and route sets. Let

A_{j r} \geq 0

be the integer amount of resource

j

required by a route-

r

call,

C_{j}

its capacity, and

S = {n \in Z_{+}^{R} : A n \leq C} .

(1)

Under fixed routing, each arriving call has a predetermined resource requirement. Take independent Poisson processes of rates

ν_{r}

and independent holding times with exponential distribution of rates

μ_{r}

. A feasible arrival changes

n

n + e_{r}

at rate

ν_{r}

; a departure changes it to

n - e_{r}

at rate

μ_{r} n_{r}

. Put

ρ_{r} = ν_{r} / μ_{r}

and assume each route uses a finite positive-capacity resource, so the state space is finite.

The product-form stationary distribution of a loss network is

π (n) = \frac{1}{G ( C )} r \prod \frac{ρ _{r}^{n_{r}}}{n _{r} !}, G (C) = n \in S \sum r \prod \frac{ρ _{r}^{n_{r}}}{n _{r} !} .

(2)

For any feasible adjacent pair,

π (n) ν_{r} = π (n + e_{r}) μ_{r} (n_{r} + 1) .

(3)

These detailed balance equations prove stationarity and make the process a reversible Markov chain. Equivalently, Independent random variables with Poisson distributions of means

ρ_{r}

are conditioned on satisfying the joint capacity constraints. The conditioning makes resource occupancies dependent even though the unconstrained counts are independent.

By Poisson arrivals see time averages, a route-

r

arrival sees acceptance probability

G (C - A_{r}) / G (C)

, interpreting the numerator as zero for a negative capacity. Hence its blocking probability is

1 - G (C - A_{r}) / G (C)

and the expected value of its number in service is

ρ_{r} G (C - A_{r}) / G (C)

. This connects a stationary occupancy law to observable rejection and carried traffic.

The insensitivity of loss networks extends this occupancy formula to independent general holding-time distributions with the same expected values, under the usual fixed resource requirements and admission rule. Counts alone then need not be a Markov chain; residual holding times belong in a Markov description. The invariant occupancy formula survives. This is useful because detailed call-duration distributions can be difficult to estimate, while their means are much easier to measure.

 Read the full article

 New to topics? Read the docs here!