1. Network Design and Analysis-----Wang Wenjie Queueing System IV: 1 © Graduate University, Chinese academy of Sciences. Network Design and Analysis Wang Wenjie Wangwj@gucas.ac.cn

2. Network Design and Analysis-----Wang Wenjie Queueing System IV: 2 © Graduate University, Chinese academy of Sciences. Queueing System IV Discrete-Time Queuing Systems

3. Network Design and Analysis-----Wang Wenjie Queueing System IV: 3 © Graduate University, Chinese academy of Sciences. 1. Discrete-Time Queuing Systems 1. 1. Motivation 1. 2. The Bernoulli Process 1. 3. Geo/Geo/1 Systems 1.4. Geo/Geo/1/N Systems 1. 5. Simple ATM Queuing Systems

4. Network Design and Analysis-----Wang Wenjie Queueing System IV: 4 © Graduate University, Chinese academy of Sciences. 1.1 Motivation Some queuing systems operate on a slotted time basis: DT Systems

5. Network Design and Analysis-----Wang Wenjie Queueing System IV: 5 © Graduate University, Chinese academy of Sciences. Model All events (arrivals, departures) must occur at integer multiples of the slot time T S Implication: all service times must be multiples of T S For convenience, often use normalized time Example: ATM – Fixed-size cells imply all service times equal 1 time slot (on a given link)

6. Network Design and Analysis-----Wang Wenjie Queueing System IV: 6 © Graduate University, Chinese academy of Sciences. Discrete-Time Markov Chain Let  j (n) = P[ X(n) = j ], j= 0, 1, … This denotes the probability of being in state j at time n, so On Off 1-    1- 

7. Network Design and Analysis-----Wang Wenjie Queueing System IV: 7 © Graduate University, Chinese academy of Sciences. Transition Probability Matrix

8. Network Design and Analysis-----Wang Wenjie Queueing System IV: 8 © Graduate University, Chinese academy of Sciences. Properties of P A square matrix whose dimension is the size of the state space Every row adds up to 1  (n+m) =  (n) P m Therefore, if we know the initial state pmf  (0), we can get the state pmf at any time m. If DTMC is stationary, in steady-state:  P

9. Network Design and Analysis-----Wang Wenjie Queueing System IV: 9 © Graduate University, Chinese academy of Sciences. Exercise 1.1 An urn initially contains 5 black balls and 5 white balls. The following experiment is repeated indefinitely: A ball is drawn from the urn If the ball is white it is put back in the urn If the ball is black it is left out X(n) is the number of black balls after n draws from the urn

10. Network Design and Analysis-----Wang Wenjie Queueing System IV: 10 © Graduate University, Chinese academy of Sciences. Exercise 1.1(Cont’d) 1.Draw the Markov chain and find the transition probabilities. 2.Find the matrix P. 3.Find the probability that there are 4 black balls in the urn after 2 draws.

11. Network Design and Analysis-----Wang Wenjie Queueing System IV: 11 © Graduate University, Chinese academy of Sciences. 1.2. The Bernoulli Process Discrete-time analogous to Poisson process – P[1 arrival during a time slot] = a – P[0 arrivals during a time slot] = 1-a Mean # of arrivals during a time slot = ____ Motivation: synchronous high-speed packet switches where at most one packet can be transmitted over a link during a slot Similarities to the Poisson process Memoryless Multiplexing and demultiplexing of a Bernoulli process still result in Bernoulli processes.

12. Network Design and Analysis-----Wang Wenjie Queueing System IV: 12 © Graduate University, Chinese academy of Sciences. Geometric Distribution (1/2) P[next arrival occurs within k time slots] = P[k-1 empty slots] P[arrival] = (1-p) k-1 p Geometric distribution Poisson process  Exponential Interarrival Time Bernoulli process  Geometric Interarrival Time

13. Network Design and Analysis-----Wang Wenjie Queueing System IV: 13 © Graduate University, Chinese academy of Sciences. Geometric Distribution (2/2) Geometric distribution has the memoryless property Mean : 1/p Variance : (1-p)/p2

14. Network Design and Analysis-----Wang Wenjie Queueing System IV: 14 © Graduate University, Chinese academy of Sciences. Binomial Distribution N slots with i arrivals in some sequence : P(sequence)=p i (1-p) N-i There are i arrivals in N slots for some value of p, the binomial distribution is : b(i,N, p)= C i N p i (1-p) N-i Mean : Np

15. Network Design and Analysis-----Wang Wenjie Queueing System IV: 15 © Graduate University, Chinese academy of Sciences. 1.3. Geo/Geo/1 Systems Bernoulli arrivals, geometric service times (DT analog to M/M/1)

16. Network Design and Analysis-----Wang Wenjie Queueing System IV: 16 © Graduate University, Chinese academy of Sciences. Model Let a = P[arrival] Let s = P[service completion] (departure) Note that this is a discrete time Markov chain, so these are transition probabilities.NOT transition rates.

17. Network Design and Analysis-----Wang Wenjie Queueing System IV: 17 © Graduate University, Chinese academy of Sciences. Local Balance Equations Analogous to  in M/M/1 system

18. Network Design and Analysis-----Wang Wenjie Queueing System IV: 18 © Graduate University, Chinese academy of Sciences. State Probabilities

19. Network Design and Analysis-----Wang Wenjie Queueing System IV: 19 © Graduate University, Chinese academy of Sciences. Average # of Customers If 0< r <1 then:

20. Network Design and Analysis-----Wang Wenjie Queueing System IV: 20 © Graduate University, Chinese academy of Sciences. Traffic Intensity

21. Network Design and Analysis-----Wang Wenjie Queueing System IV: 21 © Graduate University, Chinese academy of Sciences. Mean Delay System throughput (customers/slot time) = a = Apply Little’s law

22. Network Design and Analysis-----Wang Wenjie Queueing System IV: 22 © Graduate University, Chinese academy of Sciences. 1.4. Geo/Geo/1/N Systems Similar to Geo/Geo/1, except for state N

23. Network Design and Analysis-----Wang Wenjie Queueing System IV: 23 © Graduate University, Chinese academy of Sciences. Local Balance Equations As before, local balance yields However Now:

24. Network Design and Analysis-----Wang Wenjie Queueing System IV: 24 © Graduate University, Chinese academy of Sciences. State Probabilities

25. Network Design and Analysis-----Wang Wenjie Queueing System IV: 25 © Graduate University, Chinese academy of Sciences. Delay P[block] = P[N in system] =  N So, throughput is a(1-  N )

26. Network Design and Analysis-----Wang Wenjie Queueing System IV: 26 © Graduate University, Chinese academy of Sciences. Simple ATM Queuing System(1)

27. Network Design and Analysis-----Wang Wenjie Queueing System IV: 27 © Graduate University, Chinese academy of Sciences. Simple ATM Queuing System(2) Deterministic service time of one time slot –Normalized to cell transmission time Always a service completion if non-empty –Server is never idle if a job (cell) is waiting Simple Bernoulli is not interesting since there is no queuing Of interest are systems where there can be up to M > 1 arrivals per time slot, e.g., from M different input sources

28. Network Design and Analysis-----Wang Wenjie Queueing System IV: 28 © Graduate University, Chinese academy of Sciences. M-Geo/D=1/1/N Systems (1)

29. Network Design and Analysis-----Wang Wenjie Queueing System IV: 29 © Graduate University, Chinese academy of Sciences. M-Geo/D=1/1/N Systems (2)

30. Network Design and Analysis-----Wang Wenjie Queueing System IV: 30 © Graduate University, Chinese academy of Sciences. M-Geo/D=1/1/N Systems (3) Transition Matrix

31. Network Design and Analysis-----Wang Wenjie Queueing System IV: 31 © Graduate University, Chinese academy of Sciences. Global Balance Equations

32. Network Design and Analysis-----Wang Wenjie Queueing System IV: 32 © Graduate University, Chinese academy of Sciences. Fast Computer Algorithm

33. Network Design and Analysis-----Wang Wenjie Queueing System IV: 33 © Graduate University, Chinese academy of Sciences. Blocking Probability

34. Network Design and Analysis-----Wang Wenjie Queueing System IV: 34 © Graduate University, Chinese academy of Sciences. M-Geo/D=1/1/ 

35. Network Design and Analysis-----Wang Wenjie Queueing System IV: 35 © Graduate University, Chinese academy of Sciences. Summary Queuing Theory(1)

36. Network Design and Analysis-----Wang Wenjie Queueing System IV: 36 © Graduate University, Chinese academy of Sciences. Summary Queuing Theory(2)

37. Network Design and Analysis-----Wang Wenjie Queueing System IV: 37 © Graduate University, Chinese academy of Sciences. Summary Queuing Theory(3)