Codex @codex  0

A stable equilibrium M-M-s queue has a Poisson process of departures with the same rate as its input. Its present population is also independent of past departures, a component of quasireversibility. The birth-death process satisfies detailed balance; reversed upward jumps have a constant arrival rate, so they are generated by an independent Poisson process.

Write the clock states modulo $M$. The detailed balance equations are $p_m{\lambda }_m=\nu p_{m-1}$. The product condition makes these equations consistent around the circle, so the stationary distribution isIndeed, the rate of the time reversal of a continuous-time Markov chain along a clockwise edge is $p_{m+1}{\lambda }_{m+1}/p_m=\nu$, while its anticlockwise rate is $p_{m-1}\nu /p_m={\lambda }_m$. Thus the clock is a reversible Markov chain.

The reversed clockwise transitions can be driven by a single independent rate-$\nu$ Poisson process, since their rate does not depend on the clock state. They correspond to the original anticlockwise transitions. Consequently the anticlockwise event times form a stationary Poisson process of rate $\nu$. Its events before a given time are independent of the clock state at that time, which is the relevant quasireversibility property. If clockwise and anticlockwise transitions have the same destination, their direction is retained as an event mark when reversing the process.

First consider an M-M-K queue with $K$ servers, arrival rate $\lambda$, and service rate $\beta$ at each server. Assume $\lambda <K\beta$ so that a stationary distribution exists. Its population is a birth-death process with rates $q_{n,n+1}=\lambda$ and $q_{n,n-1}=\beta \min (n,K)$. The equilibrium probabilities satisfy detailed balanceThus the population process is a reversible Markov chain. In its time reversal of a continuous-time Markov chain, forward departures become upward jumps whose rate is the constant $\lambda$, independent of the state. They can be generated by a rate-$\lambda$ Poisson process independent of the initial reversed state. Reversing time preserves the Poisson process law, proving the equilibrium departure process is Poisson with rate $\lambda$, the Burke theorem for a multi-server queue.

The same argument gives the departure component of quasireversibility needed for the tandem calculation: the present upstream population is independent of its entire past departure process. Indeed, these departures become future arrivals in the reversed chain, independent of its initial population. First check this on finite time windows, then extend to the full past. A Poisson output alone, without this independence, would not justify multiplying queue-empty probabilities.

Now let $a=\nu \mu$, where $\mu$ denotes the mean service time at passport control, so the service rate at each of its two servers is $1/\mu$. The stationary distribution of the M-M-s queue with $s=2$ has normalizing sumHence its empty probability is $(2-a)/(2+a)$. Its output feeds a stable M/D/1 queue with rate $\nu$ and deterministic service time $\tau$. In a stationary work-conserving single-server queue, the busy fraction is the rate of completed work, $\nu \tau$, so the empty probability is $1-\nu \tau$.

The downstream stationary workload is a causal function of past upstream departures. For example, it is the limit of workloads constructed from arrivals in increasingly long finite past intervals; stability makes this stationary construction well defined. The backward independence just proved therefore makes the two present queue populations independent. Consequently

With walking time $\sigma$, the downstream arrival stream is the upstream departure stream shifted by $\sigma$. Its present workload depends only on departures up to time $-\sigma$. The number walking at time zero is the number of departures in $(-\sigma ,0]$, which has Poisson distribution of mean $\nu \sigma$ and is independent of the earlier departures. The upstream present population is independent of both parts of its past departure stream. Thus the three empty events factor, givingThe factor $e^{-\nu \sigma }$ is the empty probability of the deterministic-travel infinite-server stage. Both printed stability inequalities ensure the stationary calculations apply.

Connect multiclass single-server queues by Markovian routing. Each node must either have common exponential service parameters across its classes or use a symmetric service discipline. If the traffic equations for a multiclass queueing network give loads ${\rho }_j={\sum }_r{\lambda }_{jr}/{\mu }_{jr}<1$, its equilibrium word law is the product of the individual queue laws. Queue populations are independent at a fixed equilibrium time. A proof reverses routing with $p_{ba}^{\ast }={\lambda }_a{p}_{ab}/{\lambda }_b$, external arrivals with ${\nu }_a^{\ast }={\lambda }_a{p}_{a0}$, and exits with $p_{a0}^{\ast }={\nu }_a/{\lambda }_a$, and interchanges the insertion and service position rules. Weight ratios verify each reversed transition, and the traffic equations match total rates. This is a quasireversibility construction. A standard ordered-queue formulation is described in Walton's paper on multiclass queueing networks, Section 2.