Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 28L (Applied Probability)

The jump chain moves away from zero with probability $2/3$ and towards zero with probability $1/3$ on either half-line. Starting from $1$, the probability that this biased walk ever hits zero is $(1/3)/(2/3)=1/2$, and the same holds starting from $-1$. Thus the return probability to zero is $1/2<1$. The jump chain, and hence $X$, is transient.

It is also explosive. The transient jump chain visits zero only finitely often. After its last visit it stays on one half-line, and the strong law for its increments givesalmost surely. In particular, eventually $|Y_n|\ge n/6$. Conditional on the jump-chain path, the mean total remaining holding time isThe sum of the actual nonnegative holding times is therefore finite almost surely, so infinitely many jumps occur in finite time.

Despite this, the chain has an invariant distribution in the continuous-time balance-equation sense $\pi Q=0$. Detailed balance across the edge from zero to one gives ${\pi }_1={\pi }_0/2$, and on each half-line it givesBy symmetry and normalization,This example also shows why, for an explosive chain, existence of an invariant distribution need not imply positive recurrence.

False. Let the jump-chain state space be ${\mathbb{Z}}_{+}$, withIt is irreducible and has invariant distributionso it is positive recurrent.

Give the continuous-time chain holding ratesand set $q_{ij}=q_i{p}_{ij}$. The rates are bounded, so the chain is nonexplosive. A return cycle from zero jumps to $i$ with probability $2^{-i}$ and then has mean holding time $2^i$ there. Consequently its mean return time is at leastThus the continuous-time chain is not positive recurrent.

Equivalently, the jump-chain invariant distribution would induce the continuous-time invariant measure ${\mu }_i/q_i$, but this has infinite total mass.

This statement is also false. Let the jump chain be simple symmetric random walk on $\mathbb{Z}$, which is null recurrent, and choose holding ratesThus $q_{i,i+1}=q_{i,i-1}=q_i/2$. The counting measure ${\mu }_i=1$ is invariant for the jump chain, sois an invariant distribution for the continuous-time chain, where $C$ is the normalizing constant.

The chain is nonexplosive: its jump chain returns to zero infinitely often, and the independent holding times on those visits have rate $q_0=1$, so their sum diverges almost surely. For an irreducible nonexplosive continuous-time chain, existence of an invariant distribution is equivalent to positive recurrence. Hence $X$ is positive recurrent although its jump chain is null recurrent.

While the queue is nonempty, service completions occur at rate $\mu$. A completion reduces the queue length only when the customer leaves, which has probability $1-p$. Thinning therefore makes $L$ a birth--death chain with birth rateand effective death rateThe standard birth--death classification givesAt equality it is null recurrent.

The recurrent chain is positive recurrent precisely when the drift is strictly towards zero:The equality case is null recurrent.

Assume $\rho =\lambda /[\mu (1-p)]<1$. Detailed balance givesso normalization yieldsThis is a geometric distribution on ${\mathbb{Z}}_{+}$, and therefore

In the stable regime, $L$ is exactly the queue-length process of an $M/M/1$ queue with arrival rate $\lambda$ and effective service rate $\mu (1-p)$. By Burke theorem, when this queue is in equilibrium its external departure process is a Poisson process of rate