1. CS 4594 Broadband Intro to Queuing Theory

2. Kendall Notation Kendall notation: [Kendal 1951] A/B/c/k/m/Z A = arrival probability distribution (most often M = Poisson (multiplicative or Markov)) B = service probability distribution (most often M = Poisson (multiplicative or Markov)) c = number of servers k = maximum queue size (most often infinite) m = customer population (most often infinite) Z = type of queuing discipline (most often FCFS)

3. Performance Measures 1. Mean queue length 2. Mean waiting time 3. Mean time a job spends in the system 4. Utilization 5. Relation between arrival and service distributions

4. The Poisson Distribution POISSON POSTULATES N(t) = number of arrivals during a time interval of length t. 1. For small t, the probability of 1 arrival is proportional to t. 2. For small t, the probability of more than 1 arrival is negligible. 3. Occurrence of an arrival is independent of other arrivals and last arrival.

5. DEVELOPMENT OF THE POISSON DISTRIBUTION P(n, t) = Prob[N(t) = n] P(n, t) >= 0 P(0, 0) = 1 P(n, 0) = 0, for n > 1 P'(0, 0) = - lambda P'(1, 0) = lambda P'(n, 0) = 0, for n > 1 P(0, t) + P(1, t) +... = 1 P(n, t + s) = P(n, t)*P(0,s) + P(n-1,t)*P(1,s) +... + P(0,t)*P(n,s)

6. Differential Equations P'(n, t) = P(n, t)*P'(0,s) + P(n- 1,t)*P'(1,0) +... + P(0,t)*P'(n,0) P'(n, t) = - lambda * P(n,t) + lambda * P(n-1,t) These can be solved iteratively, starting with P(0, t).

7. For n=0, P'(0, t) = -lambda * P(0, t) leads to P(0, t) = A * exp(-lambda * t) but P(0, 0) = 1 means that A = 1 ThusP(0, t) = exp(-lambda * t)

8. For n=1 we can derive P(1, t) = (lambda * t) * exp(-lambda * t)

9. In general, the following is true: P(n, t) = (lambda * t)^n /n! * exp(-lambda * t)

10. DISTRIBUTION OF INTERARRIVAL TIMES Probability of an event occurring at time T <= t : F(t) = prob[T <= t] = P(1,t) + P(2, t) +... = 1 - P(0, t) = 1 - exp(-lambda * t)