1Network Performance...Network DesignPerformance Evaluation, and Simulation#6Section 4.7 & Section 5.7 & Appendix A2Network Performance...Network Design Problem Goal Given – QoS metric, e.g.,  Average delay Loss probability– Characterization of the traffic, e.g., Average interarrival time (arrival rate) Average holding time (message length) Design the system  Three systems will be studied:– Circuit switch: Determine the # lines System 1 M/M/S/S (M/M/S/S /∞)– Ideal router output port: Determine link capacity System 2 M/M/1 (M/M/1/∞/∞) – Real router output port: Determine link capacity and buffer size System 3 M/M/1/N (M/M/1/N /∞)3Network Performance...Network Performance Evaluation Solution methodologies: Mathematical analysis–Model this type of process as a Queueing System good for initial design Simulation techniques good for more detailed analysis 4Network Performance...Network Performance Evaluation: Elements of a Queueing SystemSystemBlockedcustomersQueueServerServerServerDepartingcustomers5Network Performance...Network Performance Evaluation: Elements of a Queueing SystemServersDelayNumber in systemNumberin QueueNumberinServers6Network Performance...Network Performance Evaluation: Specific cases for theoretical analysis Assumptions: Interarrival times are exponentially distributed  Service times are exponentially distributed– Holding time– Packet length Types of systems– One server (Stat Mux) Infinite memory Finite memory– S servers and a system size of S (Circuit Switch)7Network Performance...Network Performance Evaluation:Approach Analysis of a pure birth process to characterize arrival processes Extension to general birth/death processes to model arrivals and departures Specialization to the specific cases to find: Probability of system occupancy,  Average buffer size,  Delay,  Blocking probability Goal: Design and analyze statistical multiplexers and circuit switching systems8Network Performance...Network Performance Evaluation:Analysis of a Pure Birth ProcessOnly Births (Arrivals) Allowed01 2Kλλλλ….K = System State (number in system)-number of arrivals for 0 to t sec -number in system at time tλ = Arrival rateArrivals and no departuresGoal: Find Prob [k arrivals in a t sec interval]9Network Performance...Network Performance Evaluation:Analysis of a Pure Birth ProcessArrivalsInterarrival Time10Network Performance...Network Performance Evaluation:Analysis of a Pure Birth Process The number represents the State of the system. In networks this is usually the number in the buffer plus the number in service. The system includes the server.  The time to clock the message bits onto the transmission facility is the service time. The server is the model for the transmission facility. Goal: Find Prob [k arrivals in a t sec interval]11Network Performance...Network Performance Evaluation:Analysis of a Pure Birth Process: Assumptions Prob[ 1 arrivals in ∆ t sec ] = λ ∆ t Prob[ 0 arrivals in ∆ t sec ]= 1- λ ∆ t Number of arrivals in non-overlapping intervals of times are statistically independent random variables, i.e., Prob [ N arrivals in t, t+T AND M arrivals in t+T, t+T+τ] =Prob [ N arrivals in t, t+T]*[M arrivals in t+T, t+T+τ] 12Network Performance...Network Performance Evaluation:Statek-1kt ∆t+ t timeHow to get to state k at t+∆ t?13Network Performance...Network Performance Evaluation:Analysis Define probability of k in the system at time t = Prob[k, t] Probability of k in the system at time t+ ∆ t= Prob[k, t+ ∆ t] = Prob[(k in the system at time t and 0 arrivals in ∆ t) or (k-1 in the system at time t and 1 arrival in ∆ t)] = (1- λ ∆ t ) Prob[k,t] + λ ∆ t Prob[k-1,t]14Network Performance...Network Performance Evaluation:Analysis Rearranging terms (Prob[k, t+ ∆ t] - Prob[k,t])/ ∆ t + λProb[k,t] = λ Prob[k-1,t] Letting ∆ t --> 0 results in the following differential equation:15Network Performance...Network Performance Evaluation:Analysis For k = 0 the solution is: Prob[0,t]=  For k = 1 the solution is: Prob[1,t]=  For k = 2 the solution is: Prob[2,t]=16Network Performance...Network Performance Evaluation:Analysis In general the solution is a Poisson probability mass function of the form:17Network Performance...Network Performance Evaluation: Analysis A Possion pmf of this from has the following moments:Poisson Arrival ProcessThe number of arrivals in any T second interval follows a Poisson probability mass function.18Network Performance...Network Performance Evaluation: Interarrival Time Analysist∆ tArrivalArrivalTaProb[t<Ta<t+ ∆ t] = Prob[0 arrivals in t sec and1 arrival in ∆ t]Prob[t<Ta<t+ ∆ t] = Prob[k=0,t]Prob[k=1, ∆ t]Prob[t<Ta<t+ ∆ t] = Ta= interarrival time λ ∆19Network Performance...Network Performance Evaluation: Interarrival Time AnalysisLet ∆ t --> 0 results in the following20Network Performance...Network Performance Evaluation:Interarrival Time AnalysisMAIN RESULT:The interarrival time for a Poisson arrival process followsan exponential probability density function.21Network Performance...Network Performance Evaluation:Birth/Death Process AnalysisThe Model for the Birth/Death ProcessNow allow arrivals and departures.22Network Performance...Network Performance Evaluation:Birth/Death Process AnalysisArrivalsDepartures23Network Performance...Network Performance Evaluation: Birth/Death Process Analysis The departure process is Poisson-- Prob[ 1 departure in ∆t sec when the system is in state k ] = µk∆t Prob[ 0 departure in ∆t sec when the system is in state k ] = 1- µk∆t Number of departures in non-overlapping intervals of times are statistically independent random variables Probability[arrival AND departure in ∆t] = 024Network Performance...Network Performance Evaluation: Birth/Death Process AnalysisPoisson service processimpliesan exponential probability density function for the message length25Network Performance...Network Performance Evaluation: Birth/Death Process AnalysisTo solve for the state probabilities:Follow the procedure used for the pure birth process and use the transitions shown26Network Performance...Network Performance Evaluation: Birth/Death Process Analysis Specific queueing systems are modeled by  Setting state dependent arrival rates, λk Setting the state