Presentation on theme: "Queuing Theory For Dummies Jean-Yves Le Boudec 1."— Presentation transcript:
Queuing Theory For Dummies Jean-Yves Le Boudec 1
All You Need to Know About Queuing Theory Queuing is essential to understand the behaviour of complex computer and communication systems In depth analysis of queuing systems is hard Fortunately, the most important results are easy We will study this topic in two modules 1. simple concepts (this module) 2. queuing networks (later) 2
1. Deterministic Queuing Easy but powerful Applies to deterministic and transient analysis Example: playback buffer sizing 3
Solution of Playback Delay Pb 5 A(t) A’(t)D(t) time bits d(0)d(0) - d(0) + (D1): r(t - d(0) + ) (D2): r (t - d(0) - ) d(t) A.
2. Operational Laws Intuition: Say every customer pays one Fr per minute present Payoff per customer = R Rate at which we receive money = N In average λ customers per minute, N = λ R 6
Little Again Consider a simulation where you measure R and N. You use two counters responseTimeCtr and queueLengthCtr. At end of simulation, estimate R = responseTimeCtr / NbCust N = queueLengthCtr / T where NbCust = number of customers served and T=simulation duration Both responseTimeCtr=0 and queueLengthCtr=0 initially Q: When an arrival or departure event occurs, how are both counters updated ? A: queueLengthCtr += (t new - t old ). q(t old ) where q(t old ) is the number of customers in queue just before the event. responseTimeCtr += (t new - t old ). q(t old ) thus responseTimeCtr == queueLengthCtr and thus N = R. NbCust/T ; now NbCust/T is our estimator of 7
Conclusions Queuing is essential in communication and information systems M/M/1, M/GI/1, M/G/1/PS and variants have closed forms Bottleneck analysis and worst case analysis are usually very simple and often give good insights … it remains to see queuing networks 33