Compute the means without listing every network state
A closed queueing network may have thousands of feasible population allocations even when the performance questions are simple: How long is a cycle? Which station holds the work? What throughput can a fixed population sustain? Mean-value analysis answers those questions recursively, moving from a network with n−1 customers to one with n.
Its key idea is physical as well as computational. An arriving customer in the eligible product-form model sees the queue-length distribution of the corresponding network with one fewer customer. That arrival theorem turns queue lengths into residence times; Little’s law closes the recursion.
This guide is a specialist extension of the Gold Standard of Queueing Theory. The sequence is deliberate: explain the model, calculate one case, change the context, and check what remains valid.
Find your route through this guide
| If you want to… | Start | Continue |
|---|---|---|
| understand the mechanism | Chapter 1 | Chapters 2–3 |
| work through the calculation | Chapter 4 | Chapter 5 |
| interpret evidence and diagnostics | Chapter 6 | Chapter 7 |
| transfer the model to a changed case | Chapter 8 | Chapter 9 |
| check limits, FAQs and practice | Chapter 10 | Chapters 11–12 |
Chapter contents
Foundations and setup
Calculation and evidence
1. Why a population recursion is possible
A single-class closed network contains a fixed number N of circulating customers. Let Dᵢ be the service demand at centre i per completed system cycle. Mean-value analysis starts at population zero, where every mean queue length is zero, then adds one customer at a time.
The method does not assume that centres are independent. The dependence created by the fixed population is carried through the n−1 queue lengths used at step n.
| State-space route | Mean-value route |
|---|---|
| Enumerate every allocation of N customers | Store centre means for populations 0 through N |
| Normalise product-form weights | Use the arrival theorem and Little’s law |
| Recover all state probabilities | Recover selected mean performance measures |
2. The arrival theorem
For the product-form closed networks covered by the theorem, a tagged class-r customer arriving at centre i sees the stationary network with one fewer class-r customer. This is not ordinary PASTA: the network has no external Poisson stream, and the arriving customer belongs to the conserved population.
For a single class at a load-independent single server, the mean residence time at population n is Rᵢ(n)=Dᵢ[1+Qᵢ(n−1)]. The customer receives its own service demand and waits behind the mean number already present in the reduced-population network.
Different centre types have different residence-time expressions. The compact formula is not a universal service-centre law.
3. Throughput and Little’s law close the loop
After calculating every centre residence time, the mean cycle time is R(n)=ΣᵢRᵢ(n) when Dᵢ already includes visit frequency. System throughput is X(n)=n/R(n). Mean queue length then follows from Qᵢ(n)=X(n)Rᵢ(n).
The conservation check is ΣᵢQᵢ(n)=n. If it fails beyond rounding, the units, visit ratios or centre formulas are inconsistent.
When visits Vᵢ and service times Sᵢ are stored separately, use Dᵢ=VᵢSᵢ and keep the system-throughput reference explicit.
4. Worked example: two centres and three customers
Take two single-server centres with demands D₁=1.0 s and D₂=0.5 s per system cycle. Begin with Q₁(0)=Q₂(0)=0 and recurse to N=3.
| n | R₁(n) | R₂(n) | X(n) | Q₁(n) | Q₂(n) |
|---|---|---|---|---|---|
| 1 | 1.000 | 0.500 | 0.6667 | 0.6667 | 0.3333 |
| 2 | 1.6667 | 0.6667 | 0.8571 | 1.4286 | 0.5714 |
| 3 | 2.4286 | 0.7857 | 0.9333 | 2.2667 | 0.7333 |
At n=3, the centre means add to 3. Centre 1 holds more customers because its service demand is twice centre 2’s. Throughput is below the asymptotic bottleneck capacity 1/D₁=1 cycle per second.
5. Read the recursion rather than only the final row
Each new customer increases waiting, but not equally at every centre. At centre 1, Q₁ grows from 0.6667 to 2.2667 across the three rows; centre 2 remains below one. The recursion exposes where the population accumulates.
The throughput increments shrink as n grows: 0.6667, then 0.8571, then 0.9333. More circulating work initially fills idle capacity, then increasingly becomes waiting at the bottleneck.
A final throughput without the population ladder hides this saturation story.
6. Load-independent, delay and load-dependent centres
A delay centre has effectively infinite parallel service, so a customer does not queue behind other customers and its residence time remains its demand. A load-independent single server uses the one-plus-mean-queue expression. Load-dependent centres require occupancy probabilities or specialised recursions.
| Centre | Residence-time idea | Caution |
|---|---|---|
| Single server | Demand × (1 + reduced-population mean queue) | Assumes eligible product form |
| Delay centre | Demand | No queueing term |
| Load dependent | Condition on occupancy-dependent rate | Means alone may be insufficient |
7. How convolution reaches the same network
Convolution algorithms compute normalising constants or marginal probabilities by combining station factors one centre at a time. Mean-value analysis instead advances the customer population and directly updates mean measures.
The two approaches are connected by the same product-form structure. Normalised convolution can work with probabilities and throughputs to reduce overflow risk, and hybrid methods can use the numerically convenient representation for each centre.
Agreement on a small network is a strong implementation check: direct enumeration, convolution and MVA should produce compatible means when their assumptions match.
8. Transfer check: turn centre 2 into a delay centre
Keep D₁=1.0 s and D₂=0.5 s, but let centre 2 provide infinite-server delay. Predict first: its residence time stays 0.5 s for every population, so fewer customers wait there and more of the population pressure appears at centre 1.
Re-run the recursion with R₂(n)=0.5. The method survives, but the centre formula changes. Copying the single-server expression would fabricate queueing at a centre designed not to queue.
This is a real transfer test: the population, demands and outputs remain recognisable while one modelling assumption changes the recursion.
9. Multiclass networks and population vectors
With several closed classes, the state of the recursion is a population vector rather than one integer. The arrival theorem removes one customer from the arriving class, and class-specific throughputs and queue lengths are updated over a multidimensional grid.
The number of population vectors grows quickly. Exact multiclass MVA remains far smaller than full state enumeration in many cases, but storage and computation can still become substantial.
Approximate MVA methods reduce this burden by interpolating or aggregating. Once approximation enters, conservation and benchmark checks become even more important.
10. Assumptions, limits and misconceptions
- The exact recursion depends on the relevant product-form and arrival-theorem conditions.
- Closed-network arrivals are not external Poisson arrivals, so ordinary PASTA is not the justification.
- A service demand combines visits and service time; mixing it with raw service time changes throughput units.
- Mean-value analysis returns means, not tail probabilities or the complete state distribution.
- Load-dependent, priority, blocking and synchronisation features need appropriate extensions.
- Approximate MVA should not be reported as exact merely because the recurrence looks exact.
Use simulation or a richer model when the performance decision depends on distributions, synchronisation or non-product-form behaviour.
11. Frequently asked questions
Why does the arriving customer see n−1 customers?
The tagged customer is one of the conserved population. Removing it leaves the reduced network whose stationary distribution appears in the arrival theorem.
Is MVA always faster than convolution?
Not universally. Complexity depends on classes, centres, populations and the outputs required. They are complementary product-form algorithms.
Can MVA predict a 99th-percentile delay?
Not from the standard mean recursion alone. A distributional method, bounds or simulation is needed.
12. A reproducible calculation checklist
| Check | Pass condition |
|---|---|
| Demand units | Every Dᵢ uses one cycle reference |
| Base case | All Qᵢ(0)=0 |
| Centre formula | Matches server type |
| Population | ΣQᵢ(n)=n at each row |
| Capacity | X(n) stays consistent with bottleneck limits |
| Transfer | Changed assumption is propagated through all rows |
The gold standard is not the recursion by itself. It is a transparent model, a complete population ladder and a changed-context test that shows which formula was structural and which was centre-specific.
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