Stochastic Processes · free preview
Renewal Theory in Queueing
A single-server queue is stable exactly when arrivals are slower than service, Little's formula ties average length to average wait through the arrival rate, and the Pollaczek-Khintchine formula prices the queue's dependence on service-time variability.
1. When a queue survives
When a queue survives
2. Little's formula
Little's formula
3. The M/G/1 embedded chain
The M/G/1 embedded chain
4. PASTA and the waiting price
PASTA and the waiting price
5. The stability race
A single-server queue has arrivals at rate lambda and service rate mu. What happens when lambda exceeds mu?
6. Two ways to run the till
What is the one-sentence idea behind Little's formula L equals a W?
7. The breather between busy periods
In an M/G/1 queue with lambda below mu, what does the renewal argument give for the mean busy period?
8. Poisson eyes see averages
What does PASTA say, and why does it fail for non-Poisson arrivals?
9. Variance breaks queues
The Pollaczek-Khintchine formula shows which feature of the service time drives long waits?
10. Reading the help desk numbers
Arrivals come every six minutes, services have mean five minutes and standard deviation seven. Why is the average queue wait 37 minutes?
11. A half-hour sprint
At a help desk customers arrive every six minutes on average and the mean service is five minutes. Compute the mean busy period in minutes and write the number.
12. Half an hour in line
For the same help desk with service mean 5 minutes and standard deviation 7 minutes, compute the average waiting time in queue W sub Q in minutes and write the number.
13. Seven deep on average
Using Little's formula with arrival rate one sixth per minute and average time in system W equal to 42 minutes, compute the average number of customers in the system and write the number.
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