EE8405 Chap.4 Queueing Measures and Little's Formula
Queueing Measures and Little's Formula
Queue measures become meaningful only after the system boundary and customer population are fixed. This chapter separates total number from number waiting, total delay from waiting time, offered traffic from carried traffic and arrival rate from admitted throughput. Little's formula then links matching means across the same boundary and provides consistency checks with utilization, service time and flow balance.
What this chapter covers
- 01
System boundaries
- 02
Little's formula
- 03
Waiting-room measures
- 04
Utilization
- 05
Offered and carried traffic
- 06
Loss and delay systems
Worked example · free
Reconcile a repair queue
- 3Apply Little's formula to the whole system.
- 2Subtract mean service time.
- 3Check the waiting-room and service components.
Key terms
- Carried Traffic
- Traffic actually admitted and served after any blocking.
- Utilization
- The long-run busy fraction of the available service capacity.
Queueing Measures and Little's Formula FAQ
Does Little's formula require exponential times?
No. It is a broad long-run flow identity and does not depend on exponential distributions, although the means and population must correspond to the same stable boundary.
Which rate belongs in a loss system?
For admitted-customer delay, use carried throughput rather than offered arrivals because blocked customers never cross the service-system boundary. The rate and time must describe the same customers.
What cannot Little's formula provide?
It does not determine a delay distribution, blocking probability or tail percentile. It converts a matching long-run mean number, throughput and mean time. Another model supplies distributional detail.
How do I choose between total and waiting-only boundaries?
Draw the waiting room and server as separate boxes. For the whole system, match total mean number with total delay; for the waiting room, match mean number waiting with waiting time. Both use throughput for the same admitted population. Reconcile the results by adding mean number in service and mean service time. If those differences fail, inspect whether one symbol includes service while its partner excludes it.
What is the difference between traffic intensity and throughput?
Traffic intensity is average occupied service capacity and is measured in erlangs, whereas throughput is a completed or admitted customer rate. Traffic volume accumulates holding time over an observation interval. They can be related through mean holding time but do not share units.
Write the observation interval and boundary before converting, especially in a loss system where offered traffic exceeds carried traffic and external demand exceeds admitted throughput.
Which quantities must share a boundary in Little's formula?
Mean number, throughput and mean time must describe the same customer population. Mixing offered arrivals with admitted-customer delay produces a numerically invalid identity. Draw the boundary first.
Exam move
Draw the waiting room and server, then decide whether each quantity covers one box or both. Reconcile totals through waiting plus service components. For every exercise, define customers, the system boundary and the flow crossing it before writing Little's formula. Decide whether the number includes the customer in service and whether delay includes service time.
Use matching symbols for the whole system and for the waiting room. Write arrival or throughput rate with units and decide whether it is offered or admitted. In an infinite-buffer stable queue, all external arrivals eventually enter service, but in a loss system blocked demand never crosses the service boundary. Use carried throughput for admitted-customer delay.
Practise the whole-system identity and the waiting-only identity side by side. After calculating total delay, subtract mean service time to obtain waiting and multiply by the same throughput to recover mean number waiting. For a single work-conserving server, check that total mean number equals number waiting plus utilization.
Keep the dimensions explicit: number is dimensionless, rate is customers per time, and delay is time per customer. Convert minutes and hours before multiplying. Study teletraffic vocabulary separately. Offered traffic is demand before blocking, carried traffic is the served portion, lost traffic is their difference, and traffic volume accumulates holding time over an observation interval.
Intensity is an average occupancy measured in erlangs, while throughput is a customer rate. Work an example that converts completed calls and holding minutes into volume, intensity and throughput so the units cannot be confused. State what Little's formula cannot provide. It converts matching long-run means but does not determine a blocking probability, delay distribution or percentile.
Another queueing model must supply those details. On the handwritten page, draw two boundaries and place the corresponding identities beside them, then add offered, carried and lost traffic relations. In an answer, name the population and boundary with the numerical result. A bare statement of delay or traffic is incomplete because the reader cannot tell whether service, rejection or waiting has been included.
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