EE8405 Chap.8 Service Variability and General Queues
Service Variability and General Queues
General-service queues expose information hidden by the mean. Pollaczek-Khinchine uses the service-time second moment, so variance raises FIFO waiting even when utilization is unchanged. The chapter converts mean and variance into a second moment, separates waiting from service, compares deterministic and exponential service, and keeps processor-sharing insensitivity distinct from FIFO results.
What this chapter covers
- 01
M/G/1 assumptions
- 02
Service second moments
- 03
Pollaczek-Khinchine
- 04
M/D/1 comparison
- 05
Processor sharing
- 06
Priority disciplines
Worked example · free
Compute M/G/1 delay from measured moments
- 2Convert variance and mean into the second moment.
- 2Check utilization and stability.
- 4Calculate waiting, add service and interpret variability.
Key terms
- Pollaczek-Khinchine Formula
- An M/G/1 mean-waiting identity driven by load and service-time second moment.
- Processor Sharing
- A discipline that divides server capacity among all customers currently present.
Service Variability and General Queues FAQ
Why does variance increase waiting?
Long services create residual work encountered by later arrivals. Squaring service duration makes those rare long jobs contribute disproportionately to the second moment. Mean service alone therefore misses the delay effect.
What is the deterministic-service comparison?
At equal arrival rate and mean service, M/D/1 mean waiting is half the M/M/1 value because deterministic service removes service-time variance. Keep utilization matched.
Does processor-sharing insensitivity apply to FIFO?
No. The taught processor-sharing result belongs to that discipline. FIFO M/G/1 waiting remains sensitive to the service-time second moment. State the service discipline before using either result.
Why does residual service depend on the second moment?
A random arrival is more likely to encounter a long job because that job occupies the server for more time. Squaring service duration captures the disproportionate contribution of rare long services to work seen on arrival. Compute the second moment from variance plus squared mean, keep squared time units and connect a larger value to increased FIFO waiting under otherwise matched load and discipline.
How should I validate a general-queue approximation?
Label it as an approximation, define the arrival and service variability inputs, and compare it with an exact special case such as Poisson arrivals or deterministic service. Check light-traffic and near-saturation directions, keep utilization consistent and report sensible precision. If the approximation differs materially from an exact benchmark, explain the scope before using it for capacity.
The final recommendation should identify which variability source drives the estimate.
Which service statistic enters FIFO M/G/1 waiting?
The second moment enters, not variance or standard deviation alone. Convert from the supplied mean and variance with units preserved before applying the formula. Keep squared time visible through substitution.
Exam move
Write mean, variance, second moment and utilization on separate lines. Finish by naming an operational way to reduce variability rather than merely restating the statistic. Convert every service statistic to one time unit before applying a waiting formula. If the question gives mean and standard deviation, square the standard deviation for variance and use second moment equals variance plus squared mean.
Preserve time-squared units until the formula returns time. Calculate utilization and check stability before interpreting steady-state waiting. Use Pollaczek-Khinchine as a structured derivation rather than a memorised expression: identify the arrival rate, service second moment and spare-capacity denominator, then add mean service only when total delay is requested.
Compare M/D/1 and M/M/1 with the same arrival rate and mean service to isolate variability. Deterministic service removes service-time variance, while exponential service has squared coefficient of variation one. Under the matched conditions, the waiting comparison supplies a strong numerical check. Do not transfer that ratio to arbitrary arrivals or disciplines.
Explain residual work in words: a random arrival is more likely to encounter a long job because it occupies more time, so rare long services influence waiting disproportionately. Practise two distributions with equal means and different tails, compute their second moments and predict the ordering of delay before calculating. Keep FIFO and processor sharing separate.
A processor-sharing insensitivity result belongs to that discipline and does not make FIFO waiting independent of the service distribution. State how jobs share capacity and whether the requested delay is conditional on job size. For general arrivals, label approximations honestly and enter arrival and service coefficients of variation in their proper roles.
Compare an approximation with an exact Poisson or deterministic special case and use sensible precision. On the handwritten page, place the second-moment identity beside the M/G/1 formula, list waiting versus total delay, and add limiting checks at light traffic and near saturation.
Close by naming the practical lever: reduce mean work, standardise service to reduce variance, separate long jobs, change discipline or add capacity. The statistic matters because it points to a different intervention.
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