AAE5301 Chap.6 Queueing Models and Service Capacity
Queueing Models and Service Capacity
Random demand meets limited service
Queueing converts inter-arrival and service-time information into rates, utilisation, waiting and blocking. M/M/1 has one server and waiting, M/M/c has parallel servers and waiting, and M/M/c/c has no waiting space, so a full system blocks an arrival.
This chapter uses Little's Law and Erlang-B carefully, emphasising compatible units, effective throughput and the operational handling of excess demand.
Choose the queue before choosing the formula
Identify the arrival population, service unit, number of parallel servers, waiting capacity and action taken when capacity is full. Convert all times to compatible rates and check stability where waiting is permitted.
M/M/1 and M/M/c describe systems in which accepted customers can wait, while a loss model treats a full arrival as blocked, redirected or lost. These interpretations cannot be swapped simply because the same utilisation symbol appears.
Translate performance into a service decision
Waiting probability, mean queue time and blocking probability answer different questions.
Use the throughput that matches the counted population in Little's Law, especially when some offered demand is rejected. Then compare one fewer and one more server, state the fallback for blocked demand and test whether arrival clustering or non-exponential service could change the result. Capacity is justified by the service consequence it avoids, not by utilisation alone.
What this chapter covers
- 01
Poisson arrival rate
- 02
Exponential service rate
- 03
M/M/1 utilisation and waiting
- 04
M/M/c parallel service
- 05
M/M/c/c and Erlang-B
- 06
Open and closed queueing networks
Worked example · free
Compute waiting in an M/M/1 service
- 2Convert times to hourly rates.
- 2Compute utilisation and check stability.
- 4Compute and verify waiting time.
Key terms
- Arrival Rate
- The expected number of arrivals per unit time, commonly written as lambda.
- Service Rate
- The expected number of services one server completes per unit time, commonly written as mu.
- Utilisation
- The share of service capacity demanded in a stable queueing model.
- Erlang-B
- The loss formula giving full-state blocking probability in an M/M/c/c system.
- Jackson Network
- A network of Markovian service nodes connected by fixed routing probabilities.
Queueing Models and Service Capacity FAQ
Why can waiting be long below full utilisation?
Random arrivals cluster and service times vary. Average capacity can exceed average demand while temporary bursts create a line, so stability below one does not guarantee an acceptable waiting time.
What does the final c mean in M/M/c/c?
The first c is the number of servers and the second is total system capacity. Because they are equal, no waiting space exists. An arrival finding all servers busy is blocked or redirected.
Which lambda belongs in Little's Law?
Use the throughput corresponding to the population counted. In a loss system, accepted throughput is lower than offered arrivals, so using offered lambda for completed customers would overstate the system population or time relation.
How do I check whether a waiting model is stable?
Compare offered arrival rate with total service capacity using compatible units. For one server, arrival rate must be below its service rate; for parallel identical servers, it must be below their combined rate. Near equality can still create unacceptable waiting, so stability is a minimum feasibility check rather than a service guarantee.
What should happen to a blocked request in the model?
State whether it leaves, retries, diverts to ground response or joins another system. That consequence determines the relevant cost and throughput. Calling every blocked drone request unserved can exaggerate system failure when fallback exists, while ignoring fallback delay can overstate the benefit of a small drone fleet.
How do I know whether the Markov assumptions are reasonable?
Compare observed inter-arrival and service-time patterns with the memoryless, independent assumptions behind an M/M model. Look for appointment schedules, bursts, priority classes, weather dependence or nearly fixed service durations. If those features are material, treat the simple model as a benchmark and test alternatives or simulation.
A numerically precise waiting estimate is not reliable merely because arrival and service averages were entered correctly.
Exam move
Make a rate table before any formula: inter-arrival time, lambda, service time, mu, server count and total capacity. State whether demand waits or leaves. After computing, verify direction by increasing one parameter mentally and checking that waiting or blocking moves sensibly. Draw the state range for the chosen model and label the full state.
Keep offered arrivals, accepted throughput and completed service on separate lines. Compare the proposed server count with adjacent counts and attach the change in waiting or blocking to an operational cost or response target. State where rejected or delayed demand goes next.
