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A server completes work at its own service rate, yet a stable equilibrium M/M/1 queue can release a Poisson stream at the arrival rate. The result is precise, surprising and easy to misuse when one assumption is missing.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · State the theorem carefully
Route 2 · Build the reversible argument
Route 3 · Use the output process
Route 4 · Test the limits
Route 5 · Check and continue
CHAPTER 1 OF 12 · STATE THE THEOREM CAREFULLY
1. A surprising output theorem needs exact conditions
An equilibrium M/M/1 queue receives Poisson arrivals at rate λ, serves one job at a time with independent exponential service times of rate μ, and has λ<μ. Burke’s theorem says that its departures form a Poisson process of rate λ. In the same stationary setting, the number left in the system at a chosen time is independent of the departures before that time.
The result is surprising because departures are produced by a queue. When the server is busy, completion times reflect service; when it is empty, no departure can occur until a new job arrives and finishes. The queue nevertheless transforms this dependence into Poisson output when every condition above holds.
This is not a general rule that “output resembles input.” It is an equilibrium theorem for a particular stochastic system. The useful habit is to write the assumptions beside the conclusion before using it.
CHAPTER 2 OF 12 · STATE THE THEOREM CAREFULLY
2. Reversibility explains the shape of the result
Let N(t) be the number of jobs in the system. In an M/M/1 queue, N(t) is a birth–death chain: from state n it moves to n+1 at rate λ, and from any positive state it moves to n−1 at rate μ. Its stationary probabilities are πn=(1−ρ)ρⁿ, where ρ=λ/μ.
Detailed balance gives πnλ=πn+1μ. Thus, in equilibrium, each probability flow from n to n+1 is matched by the reverse flow from n+1 to n. Watching the stationary chain backwards produces the same transition law. Forward arrivals become backward departures, so the reversed departure points have the Poisson law of arrivals. Time reversal then gives Poisson departures forward in time.
Reversibility is the mechanism. “Arrival rate equals departure rate” is only flow conservation and is far weaker.
CHAPTER 3 OF 12 · BUILD THE REVERSIBLE ARGUMENT
3. Work an equilibrium M/M/1 example
Suppose arrivals occur at λ=3 per hour and exponential service has μ=5 per hour. Then ρ=0.6 and πn=0.4(0.6)ⁿ. The long-run probability of an empty system is 0.4, while the busy fraction is 0.6.
Under stationary operation, Burke’s theorem makes departures a Poisson process of rate 3 per hour. During a four-hour interval, the departure count D has mean 12 and variance 12. The probability of no departures is e⁻¹²; the probability of exactly 12 is e⁻¹²12¹²/12!.
| Quantity | Calculation | Result |
|---|---|---|
| Traffic intensity | ρ=3/5 | 0.6 |
| Empty probability | 1−ρ | 0.4 |
| Mean departures in four hours | λ×4 | 12 |
| Departure-count variance | Poisson mean | 12 |
The numerical equality of input and output rates would also hold in many stable non-Poisson systems. The Poisson count law and independent increments are the stronger conclusions.
CHAPTER 4 OF 12 · BUILD THE REVERSIBLE ARGUMENT
4. Stationarity is doing real work
Imagine the same queue is opened empty at 9:00. In its first instant it cannot depart: it must wait for an arrival and then a service completion. The first-departure time is therefore not exponential with rate λ. The output near opening is not a stationary Poisson process, even though arrivals and services have the required distributions.
As the queue approaches equilibrium, later behaviour may be approximated by the stationary model. Whether that approximation is safe depends on the warm-up time, utilisation and measurement window. A simulation should either initialise N(0) from the stationary geometric distribution or discard a justified warm-up period before testing Burke’s conclusion.
This changed context prevents a common transfer error: using a steady-state theorem for a start-up, recovery period or daily opening cycle.
CHAPTER 5 OF 12 · BUILD THE REVERSIBLE ARGUMENT
5. Rate balance is not the Poisson property
For a stable lossless queue observed over a long period, total departures differ from admitted arrivals only by the change in backlog. Dividing by a long time gives matching average rates under mild conditions. That statement does not say that departure gaps are exponential or that counts in disjoint windows are independent.
| Claim | What it establishes | What remains unproved |
|---|---|---|
| Long-run input rate equals output rate | Flow conservation | Poisson gaps or independent increments |
| Departure counts have mean λt | First moment | Variance, tails and dependence |
| Departures are Poisson rate λ | Count law and independent increments | Validity outside the theorem’s assumptions |
| Queue is reversible in equilibrium | Same state-chain law under time reversal | Reversibility after model changes |
When evaluating data, test the complete implication rather than celebrating a matching mean.
CHAPTER 6 OF 12 · USE THE OUTPUT PROCESS
6. What the independence statement actually says
At a fixed equilibrium time t, the current queue length N(t) is independent of the departure process before t. Knowing the past departure record does not update the geometric distribution of the number now present. This quasi-reversibility feature is central to networks of queues.
It does not mean every queue variable is independent of the entire departure history. A tagged customer’s waiting time and departure time are plainly related. Nor should one infer that future departures are independent of the present state: an empty system and a crowded system imply different immediate prospects before arrivals intervene.
State the time direction and variables. Precision turns a memorable slogan into a usable theorem.
CHAPTER 7 OF 12 · USE THE OUTPUT PROCESS
7. Tandem queues inherit tractable inputs
Place a second equilibrium M/M/1 station after the first, with service rate μ₂ greater than λ. Burke’s theorem says the first station’s output is Poisson rate λ, so it supplies the correct M/M/1 input to the second station. This allows each node in an open Jackson-style tandem to be analysed locally under the network’s equilibrium conditions.
For λ=3 per hour, μ₁=5 and μ₂=4, the individual utilisations are 0.6 and 0.75. Each node has a geometric stationary queue-length distribution with its own utilisation. The second station does not receive an arbitrary smoothed stream merely because a server sits upstream; it receives Poisson output because the first node meets the theorem.
This transfer is powerful, but finite buffers, blocking, feedback, priorities or non-exponential service can break the local argument.
CHAPTER 8 OF 12 · USE THE OUTPUT PROCESS
8. Change the service law and the output changes
Replace exponential service by a fixed 12-minute service time while keeping Poisson input at 3 per hour. The system is now M/D/1. When it stays busy, successive departures are exactly 12 minutes apart, which is incompatible with exponential gaps. Empty periods add further structure. The long-run output rate can still be 3 per hour, but the process is not generally Poisson.
The memoryless exponential law makes the birth–death state description sufficient and supports time reversibility. A general service distribution carries residual-service information not represented by queue length alone.
The same caution applies to multiple servers: useful reversibility results exist for some Markovian nodes, but the single-server statement should not be copied without checking the appropriate theorem.
CHAPTER 9 OF 12 · TEST THE LIMITS
9. Operational features can invalidate the conclusion
Finite capacity rejects arrivals, so output rate is the admitted rate rather than the attempted rate. Server vacations create output gaps and bursts. Priorities reorder jobs and can alter class-specific departure processes. Abandonment removes jobs without service completions. Batch arrivals or batch service create simultaneous or clustered events.
Time-varying arrival rates remove stationarity, while state-dependent arrivals break the constant birth rate. Breakdowns add a hidden service state. Even timestamp rounding can make distinct departures appear simultaneous and falsely reject a continuous-time model.
Map the real mechanism before choosing the theorem. If a feature matters operationally, include it in the model or label the M/M/1 result as a baseline rather than a fact about the system.
CHAPTER 10 OF 12 · TEST THE LIMITS
10. Check data against the theorem’s implications
Start with provenance: distinguish attempted arrivals, admitted jobs, service completions, abandonments and retries. Estimate whether arrival gaps are consistent with a stable Poisson rate and whether service durations resemble independent exponential observations. Check the observation period for start-ups, closures and capacity changes.
For departures, compare mean and variance across equal windows, inspect gap distributions, and test serial dependence. Time-rescaling can assess a fitted nonhomogeneous Poisson alternative when rates vary. Simulate from the fitted queue with the same initialisation and measurement rules, then compare summaries that matter for the decision.
No single diagnostic proves the model. Agreement across mechanisms, timestamps, window counts and operational knowledge is more persuasive than one p-value.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers should ask
Does an empty probability of 1−ρ cause departures to slow below λ?
No. Busy periods complete at rate μ and occupy fraction ρ, so the mean completion rate is ρμ=λ. Burke’s theorem adds the much stronger Poisson structure.
Can I use the theorem during a morning start-up?
Not exactly. An empty deterministic start is not the stationary initial distribution. Treat the early period separately or justify a warm-up approximation.
Does Poisson output prove exponential service?
Not by itself. Different mechanisms can imitate some output summaries. Identify service records and test the full model.
Why is the theorem useful in networks?
Poisson output from one reversible node supplies a tractable arrival stream to the next, enabling consistent local analysis in eligible networks.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Continue through the verified queueing ecosystem
Begin with the Queueing Theory root, the Gold Standard of Education and the How X Works hub. Compare the theorem’s stationary input with Time-Varying Arrivals and Capacity That Arrives Too Late and Time-Rescaling Diagnostics for Nonhomogeneous Poisson Arrivals.
For arrival mechanisms that destroy the simple output result, use Cox Processes, Overdispersion and Latent Arrival Rates, Batch Arrivals, Event Rates and Job Counts and Retries, Attempt Rates and Unique Jobs. For capacity limits, continue to Finite Buffers, Blocking and Admitted Throughput and Server Vacations and Residual Delay.
Authoritative technical reading: the University of British Columbia’s stochastic-process notes on Burke’s theorem give the equilibrium result and reversibility argument. The University of Cambridge’s Applied Probability notes develop the theorem and its use in queueing networks. Apply their mathematics only after checking equilibrium, stability and the event definitions in the observed system.
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