Birth-death process

A birth-death process moves only between neighbouring nonnegative integer states, with state-dependent birth and death rates.

A pure birth process on ${\mathbb{N}}_0$ jumps only from $n$ to $n+1$, at a strictly positive rate ${\lambda }_n$. Its successive holding times are independent exponential variables of rates ${\lambda }_n$.

A pure birth process is nonexplosive exactly whenIts explosion time is the sum of independent exponential holding times. If the displayed series converges, that sum has finite expectation. If it diverges, the Laplace transform of the sum isforcing the sum to be infinite almost surely.

The pure birth process with rates ${\lambda }_n=\alpha n+\beta$, where $\alpha ,\beta >0$, combines linear reproduction with immigration. Starting from zero, its mean satisfiesand hence

The Yule process is the pure birth process with rate $n\lambda$ in population state $n$. Starting from one individual, it is nonexplosive because the independent holding times have divergent total almost surely.

For the Yule process starting from one individual, the generator applied to $f(n)=n$ givesand hence $\mathbb{E}{X}_t=e^{\lambda t}$.

Given $X_t=n+1$, the $n$ birth times in a Yule process have the law of the order statistics of independent variables with densityThis follows by dividing the joint holding-time density by
$\mathbb{P}(X_t=n+1)=e^{-\lambda t}(1-e^{-\lambda t}{)}^n$.

For individual birth and death rates $\lambda ,\mu$ and one initial individual, the finite-time extinction probability satisfiesWhen $\lambda \ne \mu$,

For birth rates $b_i$ and death rates $d_i$, a reversible measure satisfiesThus ${\pi }_{i+1}/{\pi }_i=b_i/d_{i+1}$, and it is an invariant distribution when the resulting measure is summable and the chain is nonexplosive.

For birth rates $b_n$, death rates $d_n$, and state probabilities $P_n$,

Since births and deaths change the state by $+1$ and $-1$,

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

The $r$th falling factorial moment is $\mathbb{E}[(N{)}_r]$, where $(N{)}_r=N(N-1)\cdots (N-r+1)$.

If $N$ is Poisson with mean $\lambda$, then $\mathbb{E}[(N{)}_r]={\lambda }^r$; in particular, $\mathbb{E}N=\lambda$ and $\mathbb{E}[N(N-1)]={\lambda }^2$.

Approximating the population law by a Poisson law replaces its first two factorial moments by $\lambda$ and ${\lambda }^2$, closing a quadratic first-moment equation as a Riccati equation for $\lambda$.

For unit jumps with birth and death rates $b(x),d(x)$, the Fokker--Planck drift and diffusion in the conventionare $u=b-d$ and $D=(b+d)/2$.

Near a stable deterministic equilibrium, replace the drift by its linearization and the diffusion coefficient by its equilibrium value. The resulting Ornstein-Uhlenbeck approximation has a stationary normal distribution.

If boundary fluxes decay sufficiently fast, two integrations by parts givewhile the diffusion term contributes zero to the first moment.

For constant birth rate $\lambda$ and death rate $\delta$ away from zero, the chain is transient when $\lambda >\delta$, null recurrent when $\lambda =\delta$, and positive recurrent when $\lambda <\delta$.

The queue length in an $M/M/1$ queue is a birth-death process with constant arrival rate $\lambda$ and service rate $\mu$ while nonempty.

If potential arrivals form a Poisson process of rate $\lambda$ and an arrival seeing $n$ customers joins with probability $p(n)$, the queue length is a birth-death process with birth rate $b_n=\lambda p(n)$ and death rate $d_n=\mu$ for $n\ge 1$. Its stationary ratios, when normalizable, are

An $M/M/1$ queue is positive recurrent exactly when its traffic intensity $\rho =\lambda /\mu$ is below one.

When $\rho <1$, the equilibrium queue length has distribution ${\pi }_n=(1-\rho ){\rho }^n$ and mean $\rho /(1-\rho )$.

At equilibrium, the departure process of a stable $M/M/1$ queue is Poisson with the same rate as the arrival process.

If a served customer returns with probability $p$, thinning service completions gives effective external departure rate $\mu (1-p)$ for the queue-length process.

In equilibrium, a Bernoulli-feedback $M/M/1$ queue has the same queue-length process as an ordinary queue with service rate $\mu (1-p)$, so its external departures form a Poisson process of rate $\lambda$.

Starting with one customer, the mean busy period of an $M/M/1$ queue iswhen $\lambda <\mu$. It is infinite when $\lambda \ge \mu$.

An $M/M/\infty$ queue has Poisson arrivals of rate $\lambda$, independent exponential service times of rate $\mu$, and infinitely many servers. Its occupancy is a birth-death process with birth rate $\lambda$ and death rate $n\mu$ in state $n$.

For every $\lambda ,\mu >0$, an $M/M/\infty$ queue is positive recurrent and has the Poisson distribution with mean $\rho =\lambda /\mu$ as its unique stationary distribution:

In stationarity, the departure process of an $M/M/\infty$ queue is a Poisson process of rate $\lambda$. By time reversibility, the departures before any fixed time are independent of the occupancy at that time.

If $X_0=n$ and $p_t=e^{-\mu t}$, thenwith independent summands. The binomial term counts surviving initial customers; Poisson thinning gives the surviving later arrivals.

Writing $\rho =\lambda /\mu$, the stationary empty probability is $e^{-\rho }$. Alternating idle periods of mean $1/\lambda$ with busy periods and applying the renewal-reward theorem gives

An $M/M/s/K$ queue has arrivals forming a Poisson process of rate $\lambda$, $s$ servers with independent service times having an exponential distribution of rate $\mu$, and total system capacity $K\ge s$. Its occupancy is the finite birth-death process on $\{0,\dots ,K\}$ with

Put $\rho =\lambda /\mu$. The unique stationary distribution of an $M/M/s/K$ queue iswhere $Z$ is the sum of the displayed weights. Adjacent terms satisfy detailed balance for a birth-death process, so the queue is reversible in equilibrium.