MAST30001 Chap.5 Queueing Systems
Queueing Systems
Queueing models connect random arrivals, service capacity and congestion. The M/M/1 system uses a Poisson arrival stream, exponential service and one server, making its queue length a birth-death process. This chapter tests stability before using stationary results, derives the geometric queue-length distribution, and links mean numbers to mean times through Little's law.
It also distinguishes waiting-only measures from quantities that include service. The key modelling choice is what the state counts and which rate is effective in each state. Formula use comes only after arrival, service, capacity and discipline assumptions are declared and a stationary regime is shown to exist.
What this chapter covers
- 01
Arrival process, service mechanism, servers, capacity and discipline
- 02
M/M/1 queue as a birth-death chain
- 03
Traffic intensity and the stability condition
- 04
Stationary queue-length distribution
- 05
Waiting-time distributions and PASTA
- 06
Little's law for system and queue measures
Worked example · free
Calculate M/M/1 congestion and delay
- 1Traffic intensity is 6/10=0.60, below one.
- 1Mean system size is 0.60/(1-0.60)=1.50.
- 2Mean total time is 1/(10-6) hour=15 minutes.
- 2Mean service is 6 minutes, so mean waiting is 15-6=9 minutes.
Key terms
- Queue Discipline
- The rule that determines which waiting customer enters service next, such as first-come first-served.
- Stable Queue
- A queue whose state distribution settles into a stationary regime under the model assumptions.
- System Size
- The number of customers waiting plus any customer currently receiving service.
- Queue Length
- The number of customers waiting for service, excluding those already in service.
- PASTA
- The property that Poisson arrivals observe the same stationary state proportions as time averages.
- Utilisation
- The long-run fraction of single-server capacity occupied, equal to traffic intensity in the stable M/M/1 model.
- Throughput
- The long-run rate at which customers complete or pass through the population being measured.
- Sojourn Time
- The total time a customer spends in the system, including waiting and service.
Queueing Systems FAQ
Why must traffic intensity be below one?
The server must complete work faster on average than work arrives. If arrival rate meets or exceeds service rate, the M/M/1 stationary weights cannot be normalised and the queue grows without a stationary regime under the unchanged assumptions.
What is the difference between L and Lq?
L is the mean number in the whole system, including a customer in service. Lq counts only customers waiting. Their time partners are total system time and pre-service waiting time, so each version of Little's law must use matching definitions.
Why does delay rise sharply near full utilisation?
For M/M/1, mean system size is rho divided by one minus rho and mean total time is one over service rate minus arrival rate. As spare capacity shrinks, these denominators approach zero, creating a steep nonlinear increase even before instability.
When does Little's law apply beyond M/M/1?
Little's law is a broad long-run conservation relation and does not require exponential service by itself. It does require consistent populations, a well-defined throughput and a stable averaging regime. Use the effective throughput when customers can be blocked, lost or abandon rather than automatically substituting the external arrival rate.
What does PASTA allow an arrival to observe?
For a Poisson arrival process independent of the system state, arrivals see the stationary state distribution in the same proportions as calendar time. The property supports arrival-view probabilities but does not mean arrivals cause no changes, and it cannot be transferred automatically to non-Poisson or state-dependent input.
How should I compare capacity changes?
Recalculate traffic intensity and delay after expressing all rates in common units. Because delay is nonlinear in spare capacity, doubling a rate does not generally halve waiting. State whether the intervention changes arrival rate, service rate, server count or service-time distribution before applying a new queue model.
Exam move
Start every queue problem by defining what the state counts and calculating traffic intensity. If the stability condition fails, stop before using stationary formulas. Keep a two-column map for system measures and queue-only measures, then rehearse the links L equals lambda times total time and Lq equals lambda times waiting time. Convert all rates to one time unit and report final times in the requested unit.
Use operational checks: average customers in service should match utilisation for one server, and total time should equal waiting plus mean service. Build a model card for each exercise containing arrival process, service law, number of servers, queue discipline, capacity and state variable. Only then select a named model.
Solve one problem from both the stationary distribution and Little's law so that the two routes cross-check. For sensitivity practice, vary the arrival rate while holding capacity fixed and explain why the delay curve steepens as spare capacity disappears. Always distinguish throughput from offered load when loss or abandonment is possible.
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