1. Performability Analysis of Wireless Cellular Networks Center for Advanced Computing and Communication Department of Electrical and Computer Engineering Duke University Durham, NC 27705 Email: kst@ee.duke.edukst@ee.duke.edu Homepage: www.ee.duke.edu/~kst Dr. Kishor S. Trivedi SPECTS2002 and SCSC2002, July, 2002

2. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

3. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

4. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Challenges in Wireless Networks Limited radio spectrum Huge demand but limited bandwidth Error-prone radio link Fading signal, less reliable link High mobility Mobility management complicated

5. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Wireless “ilities” besides performance Performability measures of the network’s ability to perform designated functions Reliability for a specified operational time Availability at any given instant Survivability Performance under failures R.A.S.-ability concerns grow. High-R.A.S. not only a selling point for equipment vendors and service providers. But, regulatory outage report required by FCC for public switched telephone networks (PSTN) may soon apply to wireless.

6. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Causes of Service Degradation Resource full Resource loss Long waiting-time Time-out Service blocking Service Interruption Loss of information Limited Resources Equipment failures Software failures Planned outages (e.g. upgrade) Human-errors in operation

7. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

8. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University The Need of Performability Modeling New technologies, services & standards need new modeling methodologies Pure performance modeling: too optimistic! Outage-and-recovery behavior not considered Pure availability modeling: too conservative! Different levels of performance not considered

9. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

10. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang Loss Pure Performance Model Let be the steady state probability for the Continuous Time Markov Chain telephone switching system : n channels call arrival process is assumed to be Poissonian with rate call holding times exponentially distributed with rate Blocking Probability: Expected number of calls in system: Desired measures of the form: _ _

11. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang Loss Pure Availability Model A telephone switching system : n channels The times to channel failure and repair are exponentially distributed with mean and, respectively. Let be the steady state probability for the CTMC Steady state unavailability: Expected number of non-failed channels: Desired measures of the form:

12. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Continuous Time Markov Chains are useful models for performance as well as availability prediction Extension of CTMC to Markov reward models make them even more useful Attach a reward rate r i to state i of CTMC X(t) is instantaneous reward rate of CTMC Markov Reward Models (MRMs)

13. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Define Y(t) the accumulated reward in the interval [0,t) Computing the expected values of these measures is easy Markov Reward Models (MRMs) (Continued)

14. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Expected instantaneous reward rate at time t: this generalizes instantaneous availability Expected steady-state reward rate: this generalizes steady-state availability Markov Reward Models (MRMs) (Continued)

15. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University MRM Measures Expected accumulated reward in interval [0,t)

16. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University MRM: Measures Expected steady-state reward rate Expected reward rate at given time Expected accumulated reward in a given interval Distribution of accumulated reward in a given interval Expected task completion time Distribution of task completion time See for more details K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition, John Wiley, 2001.

17. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

18. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang loss composite model A telephone switching system : n channels The call arrival process is assumed to be Poissonian with rate, the call holding times are exponentially distributed with rate The times to channel failure and repair are exponentially distributed with mean and, respectively The composite model is then a homogeneous CTMC

19. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang loss composite model The state (i, j ) denotes i non-failed channels and j ongoing calls in the system CTMC with (n+1)(n+2)/2 states Total call blocking probability: Example of expected reward rate in steady state State diagram for the Erlang loss composite model

20. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Total call blocking probability Blocked due to buffer full Blocked due to buffer full in degraded levels Blocked due to unavailability

21. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Problems in composite performability model Largeness: Number of states in the Markov model is rather large Automatically generate Markov reward model starting with an SRN (stochastic reward net) Use a two-level hierarchical model Stiffness: Transition rates in the Markov model range over many orders of magnitude Potential solution to both problems is a hierarchical performability model

22. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang loss hierarchical model Upper availability model Lower performance model The hierarchical performability model provides an approximation of the exact composite model Each state of pure availability model keeps track of the number of non-failed channels. Each state of the performance model represents the number of talking channels in the system Call blocking probability is computed from pure performance model and supplied as reward rates to the availability model states

23. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang loss model (cont’d) The steady-state system unavailability The blocking probability with i non- failed channels Total blocking probability Blocked due to buffer full Blocked due to unavailability (u) Blocked due to buffer full in degraded levels

24. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Erlang loss model (cont’d) Total blocking probability in the Erlang loss model Compare the exact total blocking probability with approximate result Advantages of the hierarchical Avoid largeness Avoid stiffness More intuitive No significant loss in accuracy

25. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Total blocking probability Has three summands Loss due to unavailability (pure availability model will capture this) Loss when all channels are busy (pure performance model will capture this) Loss with some channels busy and others down (degraded performance levels) Performability models captures all three types of losses Higher level, lower level model or both can be based on analytic/simulation/measurements

26. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Performability Evaluation (1) Two steps The construction of a suitable model The solution of the model Two approaches are used Combine performance and availability into a single monolithic model Hierarchical model where lower level performance model supplies reward rates to the upper level availability model

27. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Performability Evaluation (2) Measures of performability [Triv 01,Haver01] Expected steady-state reward rate Expected reward rate at given time Expected accumulated reward in a given interval Distribution of accumulate reward Expected task completion time Distribution of task completion time

28. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

29. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Handoffs in wireless cellular networks Handoff: When an MS moves across a cell boundary, the channel in the old BS is released and an idle channel is required in the new BS Hard handoff: the old radio link is broken before the new radio link is established (AMPS, GSM, DECT, D- AMPS, and PHS)

30. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Wireless Cellular System Traffic in a cell A Cell New Calls Handoff Calls From neighboring cells Common Channel Pool Call completion Handoff out To neighboring cells

31. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Performance Measures: Loss formulas or probabilities When a new call (NC) is attempted in an cell covered by a base station (BS), the NC is connected if an idle channel is available in the cell. Otherwise, the call is blocked If an idle channel exists in the target cell, the handoff call (HC) continues nearly transparently to the user. Otherwise, the HC is dropped Loss Formulas New call blocking probability, P b : Percentage of new calls rejected Handoff call dropping probability, P d : Percentage of calls forcefully terminated while crossing cells

32. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Guard Channel Scheme Handoff dropping less desirable than new call blocking! Handoff call has Higher Priority: Guard Channel Scheme GCS: g channels are reserved for handoff calls. g trade-off between P b & P d

33. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University # Modeling for cellular network with hard handoff # g g+1 n Stochastic Petri net Model of wireless hard handoff Idle-channels new Handoff-in Handoff-out Call-completion Assumptions Poisson arrival stream of new calls Poisson stream of handoff arrivals Limited number of channels: n Exponentially distributed completion time of ongoing calls Exponentially distributed cell departure time of ongoing calls G. Haring, R. Marie, R. Puigjaner and K. S. Trivedi, Loss formulae and their optimization for cellular networks, IEEE Trans. on Vehicular Technology, 50(3), 664-673, May 2001.Loss formulae and their optimization for cellular networks

34. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Markov chain model of wireless hard handoff Steady-state probability C(t): the number of busy channels at time t

35. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Loss formulas for wireless network with hard handoff Dropping probability for handoff: Blocking probability of new calls:  Notation: if we set g=0, the above expressions reduces to the classical Erlang-B loss formula

36. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Computational aspects Overflow and underflow might occur if n is large Numerically stable methods of computation are required Recursive computation of dropping probability for wireless networks Recursive computation of the blocking probability For loss formula calculator, see webpage: http://www.ee.duke.edu/~kst/wireless.html

37. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Optimization problems Optimal Number of Guard Channels Optimal Number of Channels O1: Given n, A, and A 1, determine the optimal integer value of g so as to O2: Given A and A 1, determine the optimal integer values of n and g so as to

38. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University # # g g+1 n Stochastic Petri net Model of wireless hard handoff Idle-channels new Handoff-in Handoff-out Call-completion Fixed-Point Iteration  Handoff arrival rate will be a function of new call arrival rate and call completion rates  Handoff arrival rate will have to be computed from the throughput of handoff-out transition

39. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Loss Formulas—Fixed Point Iteration A fixed point iteration scheme is applied to determine the Handoff Call arrival rates: We have theoretically proven: the given fixed point iteration is existent and unique The arrival rate of HCs=the actual throughput of departure call leaving the cell

40. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Modeling cellular systems with failure and repair (1) The object under study is a typical cellular wireless system The service area is divided into multiple cells There are n channels in the channel pool of a BS Hard handoff. g channels are reserved exclusively for handoff calls Let be the rate of Poisson arrival stream of new calls and be the rate of Poisson stream of handoff arrivals Let be the rate that an ongoing call completes service and be the rate that the mobile engaged in the call departs the cell The times to channel failure and repair are exponentially distributed with mean and, respectively.

41. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Modeling cellular systems with failure and repair (2) Upper availability model (same as that in Erlang loss model) The steady state unavailability (u)

42. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Modeling cellular systems with failure and repair (3) Lower performance model

43. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Modeling cellular systems with failure and repair (4) Solve the Markov Chain, we get pure performance indices The dropping probability The blocking probability

44. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Modeling cellular systems with failure and repair (5) Total dropping probability Total blocking probability Loss probability (now call blocking or handoff call dropping) is computed from pure performance model and supplied as reward rates to the availability model states Unavailability Buffer full Degraded buffer full

45. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov Reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

46. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Hierarchical model for APS in TDMA system (1) Each cell has N b base repeaters (BR) Each BR provides M TDM channels One control channel resides in one of the BRs Control channel down System down(!) A TDMA Cellular System Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000, Japan, Nov., 2000.

47. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Hierarchical model for APS in TDMA system (2) Platform_down The controller or the local area network connecting the base repeaters and controller going down causing the system as a whole to go down. Control_down The base repeater where the control channel resides going down causing the system as a whole to go down. Base_repeater_down Any other base repeater where the control channel does not reside going down does not cause the system as a whole to go down, but system is degraded (partially down). Failure in System

48. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Hierarchical model for APS in TDMA system (3) Upon control_down, the failed control channel is automatically switched to a channel on a working base repeater. control_down causes only system to go partially down no longer a full outage! A sys P b P d Automatic Protection Switching

49. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Hierarchical model for APS in TDMA system (4) High level: availability model Failure/recovery of base repeaters, and platform of system Lower level: performance model New call blocking probability and handoff call dropping probability for a given number of working base repeaters Combine together Lower level performance measures as reward rates assigned to states on higher level and finally the overall steady-state expected reward rate provides the total blocking (dropping) probability 2-level Decomposition

50. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Hierarchical model for APS in TDMA system (5) 0,N b 0,00,b 1,N b 1,01,b High level - availability Failure/recovery of base repeaters, and platform of system 012 n-g-1n-gn-g+1n-1n Low level - Performance New call blocking/handoff dropping A birth-death process (BDP) n = Mb - 1 channels available

51. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Performability Indices System Unavailability Overall New Call Blocking Prob. Overall Handoff Call Dropping Prob.

52. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Numerical Results (1) New Call Blocking Probability Improvement by APS Unavailability in new call blocking probability

53. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Numerical Results (2) Handoff Call Dropping Probability Improvement by APS Unavailability in handoff call dropping probability

54. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Outline Overview of wireless mobile systems Why performability modeling? Markov Reward models Erlang loss performability model Modeling cellular systems with failure Hierarchical model for APS in TDMA Conclusion

55. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Conclusions Performability: an integrated way to evaluate a real-world system Two approaches Composite models Hierarchical models CTMC and MRM models for performability study of a variety of wireless systems

56. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Future work Performability study in more general systems Call holding times generally distributed [Logothetis and Trivedi, submitted] Handoff interarrival times generally distributed [Dharmaraja et al. spects 2002] multiple control channels and corresponding fault- tolerant protection schemes other fault detection, isolation and restoration strategies in cellular system

57. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University Future work Performability study in advanced systems voice+data, multi-media wireless system Differentiated QoS services over multiple interconnected networks Packet-switched traffic: IP wireless mobile system Survivability of cellular systems

58. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University References 1.Y. Ma, J. Han and K. S. Trivedi, Composite Performance & Availability Analysis of Wireless Communication Networks, IEEE Trans. on Vehicular Technology, 50(5): 1216-1223, Sept. 2001.Composite Performance & Availability Analysis of Wireless Communication Networks 2.G. Haring, R. Marie, R. Puigjaner and K. S. Trivedi, Loss formulae and their optimization for cellular networks, IEEE Trans. on Vehicular Technology, 50(3), 664-673, May 2001.Loss formulae and their optimization for cellular networks 3. K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition, John Wiley, 2001 (especially Section 8.4.3). 4. Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000, Japan, Nov., 2000. 5.H.-R. Sun, Y. Cao, K. S. Trivedi and J. J. Han, Availability and performance evaluation for automatic protection switching in TDMA wireless system, PRDC’99, pp15--22, Dec., 1999 6.http://www.ee.duke.edu/~kst/wireless.htmlhttp://www.ee.duke.edu/~kst/wireless.html 7. B. Haverkort et al, Performability Modeling, John Wiley, 2001 8.D. Selvamuthu, D. Logothetis, and K. S. Trivedi, Performance analysis of cellular networks with generally distributed handoff interarrival times, Proc. of SPECTS2002, July 2002.

59. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University The End Thank you!