1. CanQueue September 15, 20061 Performance Modeling of Stochastic Capacity Networks Carey Williamson iCORE Chair Department of Computer Science University of Calgary

2. CanQueue September 15, 20062 Introduction  There exist many practical systems in which the system capacity varies unpredictably with time  These systems are complicated to model and understand  Main focus of this talk: Stochastic capacity networks Lots of modeling issues and questions A few answers (mostly from simulation)

3. CanQueue September 15, 20063 Some Examples  Safeway checkout line  Variable-rate servers  Load-dependent servers  Grid computing center  Priority-based reservation networks  Wireless Local Area Networks (WLANs)  Wireless media streaming scenarios  Handoffs in mobile cellular networks  “Soft capacity” cellular networks Queueing systems Loss systems

4. CanQueue September 15, 20064 Some Examples  Safeway checkout line  Variable-rate servers  Load-dependent servers  Grid computing center  Priority-based reservation networks  Wireless Local Area Networks (WLANs)  Wireless media streaming scenarios  Handoffs in mobile cellular networks  “Soft capacity” cellular networks Queueing systems Loss systems

5. CanQueue September 15, 20065 Grid Computing Example  Jobs of random sizes arrive at random times to central dispatcher, and are then sent to one of M possible computing nodes  If a computing node fails, then all jobs that are currently in progress on that node are irretrievably lost  Performance impacts: Lost work needs to be redone Increased queue delay for waiting jobs

6. CanQueue September 15, 20066 Wireless LAN (WLAN) Example  An IEEE 802.11b WLAN (“WiFi”) supports four different physical transmission rates: 1 Mbps, 2 Mbps, 5.5 Mbps, 11 Mbps  Stations can dynamically switch between these rates on a per-frame basis depending on signal strength and perceived channel error rate  Performance impacts: The presence of one low-rate station actually degrades throughput for all WLAN users [Pilosof et al. IEEE INFOCOM 2003]

7. CanQueue September 15, 20067 Cellular Network Terminology BSC PSDN BS Forward Reverse MS

8. CanQueue September 15, 20068 Cellular Handoff Example  Mobile phones communicate via a cellular base station (BS)  Movement of active users beyond the coverage area of current BS necessitates handoff to another BS  If no resources available, drop call  Possible strategies: Guard channels (static or dynamic) Power control, “soft handoff”, etc.

9. CanQueue September 15, 20069 Handoff Traffic in a Base Station Cell Site New Calls (Poisson) Channel Pool with total C channels Call completion (exponential distribution) Handoff Calls To neighbour cells Handoff Calls (non-Poisson) From neighbour cells g Guard channels (static scheme) [Dharmaraja et al. 2003] C- g (blocking possible) (dropping possible)

10. CanQueue September 15, 200610 Handoff Traffic in a Base Station Cell Site New Calls (Poisson) Channel Pool with total C channels Call completion (exponential distribution) Handoff Calls To neighbour cells Handoff Calls (non-Poisson) From neighbour cells g Guard channels (dynamic scheme) C- g (blocking possible) (dropping possible) (dropping possible!)

11. CanQueue September 15, 200611 Cellular Network Layout “hard handoff” versus “soft handoff”

12. CanQueue September 15, 200612 “Soft Capacity” Example  Problem originally motivated by research project with TELUS Mobility  Q: How many users at a time can be supported by one BS? - CLW  A: “It depends” - MW  CDMA cellular systems are typically interference-limited rather than channel limited (i.e., time varying)  Intra-cell and inter-cell interference

13. CanQueue September 15, 200613 Soft Capacity: “Cell Breathing” The effective service area expands and contracts according to the number of active users!

14. CanQueue September 15, 200614 Observation and Motivation  Networks with time-varying capacity tend to exhibit higher call blocking rates and higher outage (dropping) probabilities than regular networks  Investigating performance in such systems requires consideration of the traffic process as well as the capacity variation process (and interactions between these two processes)

15. CanQueue September 15, 200615 Research Questions  What are the performance characteristics observed in stochastic capacity networks?  How sensitive are the results to the parameters of the stochastic capacity variation process?  Can one develop an “effective capacity” model for such networks?

16. CanQueue September 15, 200616 Background: Erlang Blocking Formula  The Erlang B formula expresses the relationship between call blocking, offered load, and the number of channels in a circuit-based network

17. CanQueue September 15, 200617 Circuit-Switched Network Model Capacity for C Calls

18. CanQueue September 15, 200618 Markov Chain Model State 0 State 1 State N Call arrival process: Poisson Call holding time distribution: Exponential Blocking state

19. CanQueue September 15, 200619 Erlang B Results 2%

20. CanQueue September 15, 200620 Erlang B Model Summary Capacity C Offered Load Blocking Probability p

21. CanQueue September 15, 200621 Our Goal: Effective Capacity Model Equivalent Capacity Offered Load Blocking Probability p Dropping Probability d Dropping Policy

22. CanQueue September 15, 200622 Modeling Methodology Overview Simulation Approach Analytic Approach System Model Capacity Model Traffic Model

23. CanQueue September 15, 200623 Traffic Model State 0 State 1 State N Arrival process: Poisson, Self-similar Holding time: Exponential, Pareto

24. CanQueue September 15, 200624 Traffic and Capacity Example Traffic Arrival and Departure Process (Point Process) t Fixed Capacity C = 10 Fixed Capacity C = 4 Fixed Capacity C = 5 Traffic Occupancy Process (Counting Process) Stochastic Capacity

25. CanQueue September 15, 200625 Stochastic Capacity Example

26. CanQueue September 15, 200626 Stochastic Capacity Terminology “High variance” “Low variance”

27. CanQueue September 15, 200627 Stochastic Capacity Terminology “High frequency” “Low frequency”

28. CanQueue September 15, 200628 Stochastic Capacity Terminology “Correlated” “Uncorrelated”

29. CanQueue September 15, 200629 Stochastic Capacity Model HH LL High value Low value Medium value Value process {Ci} Timing process {ti}

30. CanQueue September 15, 200630 Effective Capacity State 0 State 1 State N HH LL High value Low value Medium value + Effects of Capacity Value process Effects of Capacity Timing process Effect of Correlations Interactions between Traffic and Capacity

31. CanQueue September 15, 200631 Full Model Structure Capacity Variation Traffic Process Blocking States Dropping Transitions

32. CanQueue September 15, 200632 Markov Chain Model for C State 0 State 1 State 3 State 2 State C 3 C

33. CanQueue September 15, 200633 Markov Chain Model for C and C-1 State 0 State 1 State C-1 State 2 (C-1) State 0 State 1 State 3 State 2 State C 3 C

34. CanQueue September 15, 200634 Parameters in Simulations ParameterLevel Network Traffic Call arrival rate (per sec)1.0 Mean holding time (sec)30 Network Capacity (calls) Mean30, 40, 50 Standard Deviation2, 5, 10 Mean Time Between Capacity Changes (sec) 10, 15, 30, 60, 120 Hurst Parameter H (for LRD model) 0.5, 0.7, 0.9

35. CanQueue September 15, 200635 Results and Observations (Preview)  Factors that matter: Mean of capacity value process Variance of capacity value process Correlation of capacity value process Frequency of capacity timing process Choice of call dropping policy used Relative time scales of joint processes  Factors that don’t matter: Distribution for capacity timing process

36. CanQueue September 15, 200636 Effect of Capacity Value Mean Large capacity C = 50 (60% load) Medium capacity C = 40 (75% load) Small capacity C = 30 (100% load)

37. CanQueue September 15, 200637 Effect of Capacity Value Variance Medium variance (75% load) High variance (75% load) Low variance (75% load)

38. CanQueue September 15, 200638 Effect of Capacity Correlation Uncorrelated Correlated

39. CanQueue September 15, 200639 Effect of Capacity Timing Process

40. CanQueue September 15, 200640 Effect of Call Dropping Policy (1 of 2)

41. CanQueue September 15, 200641 Effect of Call Dropping Policy (2 of 2)

42. CanQueue September 15, 200642 Effect of Relative Time Scale R = E[call arrivals/capacity change]

43. CanQueue September 15, 200643 Results and Observations (Recap)  Factors that matter: Mean of capacity value process Variance of capacity value process Correlation of capacity value process Frequency of capacity timing process Choice of call dropping policy used Relative time scales of joint processes  Factors that don’t matter: Distribution for capacity timing process

44. CanQueue September 15, 200644 Summary and Conclusion  Studied call-level performance in a network with stochastic capacity variation  Shows influences from the properties of the stochastic capacity variation process  Shows that mean and variance of capacity process have the largest impact, as do the correlation structure and timing  Shows impact of interactions between traffic and capacity processes  One step closer to our goal, but the hard part is still ahead!

45. CanQueue September 15, 200645 Our Goal: Effective Capacity Model Equivalent Capacity Offered Load Blocking Probability p Dropping Probability d Dropping Policy

46. CanQueue September 15, 200646 References  H. Sun and C. Williamson, “Simulation Evaluation of Call Dropping Policies for Stochastic Capacity Networks”, Proceedings of SCS SPECTS 2005, Philadelphia, PA, pp. 327-336, July 2005.  H. Sun and C. Williamson, “On Effective Capacity in Time-Varying Wireless Networks”, Proceedings of SCS SPECTS 2006, Calgary, AB, July 2006.  H. Sun, Q. Wu, and C. Williamson, “Impact of Stochastic Traffic Characteristics on Effective Capacity in CDMA Networks”, to appear, Proceedings of P2MNet, Tampa, FL, Nov. 2006.  H. Sun and C. Williamson, “On the Role of Call Dropping Controls in Stochastic Capacity Networks”, submitted for publication, 2006.

47. CanQueue September 15, 200647 Related Work  S. Dharmaraja, K. Trivedi, and D. Logothetis, “Performance Modelling of Wireless Networks with Generally Distributed Hand-off Interarrival Times”, Computer Communications, Vol. 26, No. 15, pp. 1747-1755, 2003.  V. Gupta, M. Harchol-Balter, A. Scheller-Wolf, and U. Yechiali, “Fundamental Characteristics of Queues with Fluctuating Load”, Proceedings of ACM SIGMETRICS 2006, St. Malo, France, June 2006.  G. Haring, R. Marie, R. Puigjaner, and K. Trivedi, “Loss Formulae and Optimization for Cellular Networks”, IEEE Transactions on Vehicular Technology, Vol. 50, No. 3, pp. 664-673, 2001.  B. Haverkort, R. Marie, R. Gerardo, and K. Trivedi, Performability Modeling: Techniques and Tools, 2001.

48. CanQueue September 15, 200648 Thanks!  Questions?  Credits: Hongxia Sun Jingxiang Luo Qian Wu S. Dharmaraja  For more information: Email carey@cpsc.ucalgary.cacarey@cpsc.ucalgary.ca http://www.cpsc.ucalgary.ca/~carey