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## Performability Analysis of Wireless Cellular Networks

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**Performability Analysis of Wireless Cellular Networks**Dr. Kishor S. Trivedi SPECTS2002 and SCSC2002, July, 2002 Center for Advanced Computing and Communication Department of Electrical and Computer Engineering Duke University Durham, NC 27705 Email: kst@ee.duke.edu Homepage: www.ee.duke.edu/~kst**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 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 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 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**for a specified operational time**Reliability Availability at any given instant Performance under failures Survivability Wireless “ilities” besides performance Performability measures of the network’s ability to perform designated functions 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. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Causes of Service Degradation**Long waiting-time Time-out Service blocking Limited Resources Resource full Equipment failures Software failures Planned outages (e.g. upgrade) Human-errors in operation Resource loss Service Interruption Loss of information 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 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 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 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Erlang Loss Pure Performance Model**• telephone switching system : n channels • call arrival process is assumed to be Poissonian with rate • call holding times exponentially distributed with rate _ _ Let be the steady state probability for the Continuous Time Markov Chain Blocking Probability: Expected number of calls in system: Desired measures of the form: 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: Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Markov Reward Models (MRMs)**• 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 ri to state i of CTMC X(t) is instantaneous reward rate of CTMC Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Markov Reward Models (MRMs) (Continued)**• Define Y(t) the accumulated reward in the interval [0,t) • Computing the expected values of these measures is easy Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Markov Reward Models (MRMs) (Continued)**• Expected instantaneous reward rate at time t: this generalizes instantaneous availability • Expected steady-state reward rate: this generalizes steady-state availability Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**MRM Measures**• Expected accumulated reward in interval [0,t) 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. 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 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 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 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Total call blocking probability**Blocked due to buffer full in degraded levels Blocked due to buffer full Blocked due to unavailability 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 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Erlang loss hierarchical model**The hierarchical performability model provides an approximation of the exact composite model • Upper availability model • Lower performance 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 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 (u) Blocked due to buffer full Blocked due to buffer full in degraded levels (u) Blocked due to unavailability Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Erlang loss model (cont’d)**Compare the exact total blocking probability with approximate result • Advantages of the hierarchical • Avoid largeness • Avoid stiffness • More intuitive • No significant loss in accuracy Total blocking probability in the Erlang loss model 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 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 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 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 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) Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**New Calls**Call completion Handoff Calls From neighboring cells Handoff out To neighboring cells Wireless Cellular System Traffic in a cell Common Channel Pool A Cell 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, Pb:Percentage of new calls rejected • Handoff call dropping probability, Pd : Percentage of calls forcefully terminated while crossing cells Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Handoff call has Higher Priority: Guard Channel Scheme**GCS: g channels are reserved for handoff calls. g trade-off between Pb & Pd Guard Channel Scheme Handoff dropping less desirable than new call blocking! Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Modeling for cellular network with hard handoff**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. • 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 Idle-channels n Assumptions g+1 g # Call-completion new # Handoff-out Handoff-in Stochastic Petri net Model of wireless hard handoff Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Markov chain model of wireless hard handoff**C(t):the number of busy channels at time t Steady-state probability 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 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 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 A1, determine the optimal integer value of g so as to O2: Given A and A1, determine the optimal integer values of n and g so as to Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**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 Idle-channels n g+1 g # Call-completion new # Handoff-out Handoff-in Stochastic Petri net Model of wireless hard handoff 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 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. 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) 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 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 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 Buffer full Unavailability Degraded buffer full 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 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Control channel down**System down(!) Hierarchical model for APS in TDMA system (1) A TDMA Cellular System • Each cell has Nb base repeaters (BR) • Each BR provides M TDM channels • One control channel resides in one of the BRs Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000, Japan, Nov., 2000. Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Hierarchical model for APS in TDMA system (2)**Failure in System • 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). Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**control_down causes only system to go partially down**no longer a full outage! Hierarchical model for APS in TDMA system (3) Automatic Protection Switching Upon control_down, the failed control channel is automatically switched to a channel on a working base repeater. Asys Pb Pd Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**Hierarchical model for APS in TDMA system (4)**2-level Decomposition • 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 asreward ratesassigned to states on higher level and finally the overall steady-state expected reward rate provides the total blocking (dropping) probability Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University**n-g-1**n-g n-g+1 n-1 n 0 1 2 Low level - Performance New call blocking/handoff dropping A birth-death process (BDP) n = Mb - 1 channels available Hierarchical model for APS in TDMA system (5) High level - availability Failure/recovery of base repeaters, and platform of system 0,Nb 0,b 0,0 1,Nb 1,b 1,0 Center for Advanced Computer and Communication Department of Electrical and Computer Engineering, Duke University