CIVL2700 · Transport Systems
Stochastic Queueing (M/M/1, M/D/1, M/M/1/N)
This Week-6 topic of CIVL2700 Transport Systems at the University of Sydney models booths, gates and freeway on-ramps as single-server queues with random arrivals and service. You learn to compute the traffic intensity ρ = λ/μ and the steady-state averages L, Lq, W and Wq for the M/M/1 (Poisson arrivals, exponential service), M/D/1 (constant service) and M/M/1/N (finite-capacity) models. It builds directly on the deterministic D/D/1 queues of Week 5 and feeds the traffic-engineering and intersection topics later in the unit.
What this chapter covers
- 01Kendall notation A/B/c: M = Markovian (Poisson arrivals / exponential service), D = deterministic, /N = finite capacity
- 02Traffic intensity (utilisation) ρ = λ/μ, dimensionless; stability needs ρ < 1 for infinite queues
- 03M/M/1 state probabilities P0 = 1 − ρ and Pn = ρn(1 − ρ)
- 04M/M/1 averages: L = ρ/(1 − ρ), Lq = ρ²/(1 − ρ), W = 1/(μ − λ), Wq = ρ/(μ − λ)
- 05Little's Law L = λW and Lq = λWq, plus the bridges L = Lq + ρ and W = Wq + 1/μ
- 06M/D/1 constant service halves the queue term: Lq = ρ²/[2(1 − ρ)]
- 07M/M/1/N finite capacity, blocking probability PN, and effective arrival rate λ(1 − PN)
- 08Sizing applications: storage-lane length (round up), on-ramp spillback and booth utilisation
- 09Unit discipline: keep λ and μ on the same base; report W and Wq in seconds or minutes
M/M/1 single service window (rho = 0.75)
- +1Traffic intensity: rho = lambda/mu = 360/480 = 0.75 (< 1, so the queue is stable).
- +1Mean number in system: L = rho/(1 - rho) = 0.75/0.25 = 3 veh.
- +1Mean number in queue: L_q = rho^2/(1 - rho) = 0.5625/0.25 = 2.25 veh (L - L_q = 0.75 = rho, as required).
- +1Times: W = 1/(mu - lambda) = 1/(480 - 360) = 1/120 h = 30 s; W_q = rho/(mu - lambda) = 0.75/120 h = 22.5 s (W - W_q = 7.5 s = 1/mu).
- +1Little's Law check: lambda*W = 360 x (1/120) = 3 = L, and lambda*W_q = 360 x 0.00625 = 2.25 = L_q. Consistent.
Key terms
- Traffic intensity (rho)
- The utilisation rho = lambda/mu, the fraction of time the single server is busy. Dimensionless; an infinite queue is stable only when rho < 1.
- M/M/1
- A single-server queue with Markovian (Poisson) arrivals at rate lambda and exponential service at rate mu — the baseline stochastic queue for a booth or gate.
- M/D/1
- Poisson arrivals but deterministic (constant) service time 1/mu. At the same rho its queue term L_q is exactly half that of M/M/1.
- M/M/1/N
- An M/M/1 queue with finite capacity N: at most N in the system, so arrivals that find it full are blocked. Valid even for rho >= 1 because the buffer bounds the queue.
- L and L_q
- Mean number in the system (L, queue plus the one in service) and mean number waiting in the queue only (L_q). They differ by rho: L = L_q + rho.
- W and W_q
- Mean time a vehicle spends in the system (W) and waiting in the queue before service (W_q). They differ by the service time 1/mu.
- Little's Law
- L = lambda*W (and L_q = lambda*W_q): the mean number equals the arrival rate times the mean time. Holds for any stable queue and is the standard correctness check.
- Blocking probability
- In M/M/1/N, P_N is the probability the system is full; an arriving vehicle is then turned away, so the effective arrival rate is lambda(1 - P_N).
Stochastic Queueing (M/M/1, M/D/1, M/M/1/N) FAQ
When do I use M/D/1 instead of M/M/1?
Use M/D/1 when service takes a fixed, constant time per vehicle (an automated gate or a fixed-cycle booth); use M/M/1 when service times are variable (exponential). At the same rho, M/D/1 gives exactly half the queue term, so choosing M/M/1 for constant service over-predicts the queue.
Why must rho be less than 1?
For the infinite M/M/1 and M/D/1 queues, if arrivals match or exceed service (rho >= 1) the queue grows without bound and the steady-state formulae diverge. The exception is M/M/1/N: its finite capacity N caps the queue, so it stays valid even for rho >= 1, with blocked arrivals accounted for by P_N.
Can AI help me with stochastic queueing in CIVL2700?
Yes — Sia can explain the M/M/1, M/D/1 and M/M/1/N formulae step by step, check your units on rho = lambda/mu, and walk through Little's Law as a self-check on your own practice numbers. It is a study aid that explains the method; it will not sit your assessment or guarantee a mark, so always confirm final answers against your Canvas materials.
Exam move
Anchor everything on the traffic intensity rho = lambda/mu, always computing it with lambda and mu in the same units and checking rho < 1 before you trust an infinite-queue formula. Practise picking the model straight from the wording — random service is M/M/1, constant service is M/D/1, and a stated capacity N is M/M/1/N — then write the named formula, substitute, and convert W and W_q into seconds or minutes. Finish every question by running Little's Law (L = lambda*W) as a free correctness check, and remember M/D/1 halves the queue term only, that L and L_q differ by rho, and that storage lengths round up to whole vehicles. The final exam is a 2.5-hour, 40% hurdle task, so budget time in proportion to the marks on each part and confirm the exam's open- or closed-book status and exact timing on Canvas.
Working through Stochastic Queueing (M/M/1, M/D/1, M/M/1/N) in CIVL2700? Sia is AskSia’s AI Engineering tutor — ask any CIVL2700 Stochastic Queueing (M/M/1, M/D/1, M/M/1/N) question and get a clear, step-by-step explanation grounded in how CIVL2700 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
