Mm1 Cellular Networks: GSM, GPRS, and UMTS. Mm2 Security aspects of wireless networks. Mm3 Security (cntd.), Header Compression

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1 Performance & Cross-Layer Aspects by Mm1 Cellular Networks: GSM, GPRS, and UMTS Mm2 Security aspects of wireless networks Mm3 Security (cntd.), Header Compression Mm4 Performance analysis: Simulation and analytic models Mm5 Reliability aspects Page 1 Content 1. Motivation & Background Performance Analysis in Wireless Settings Review of Basic Concepts: Random Variables, Exponential Distributions, Stochastic Processes 2. Simulation Models Basics: Discrete Event Simulation Random Number Generation Output Analysis 3. Optional: Simple Analytic Models Birth-Death Processes M/M/1 Queues Circuit-Switched case: Erlang formula Packet-based traffic models 4. Summary Page 2

2 Intro: Packet-Based Transport Advantages of Packet-Based Transport (as opposed to circuit switched) Flexibility Optimal Use of Link Capacities, Multiplex-Gain for bursty traffic Drawbacks Buffering/Queueing at routers can be necessary + Delay / Jitter / Packet Loss can occur Overhead from Headers (20 Byte IPv4, 20 Byte TCP)... and it makes performance modeling harder!! queueing ν Main motivation for Performance Modeling: Network Planning Evaluation/optimization of protocols/architectures/etc. Page 3 Challenges in Packet Switched Setting Challenges in IP networks: HTTP Multiplexing of packets at nodes (L3) TCP Burstiness of IP traffic (L3-7) IP Impact of Dynamic Routing (L3) Link-Layer Performance impact of transport layer, in particular TCP (L4) Wide range of applications different traffic & QoS requirements (L5-7) Feedback: performance traffic model, e.g. for TCP traffic, adaptive applications L5-7 L4 L3 L2 Challenges in Wireless Networks: Wireless link models (channel models) MAC & LLC modeling RRM procedures Mobility models Cross layer optimization Analysis frequently with stochastic models Page 4

3 Basic concepts Probabilities Random experiment with set of possible results Ω Axiomatic definition on event set V(Ω) 0 Pr(A) 1; Pr( )=0; Pr(A B)=Pr(A)+Pr(B) if A B= [ A,B (Ω) ] Conditional probabilities: Pr(A B)=Pr(A B) / Pr(B) Random Variables (RV) Definition: X: Ω ú; Pr(X=x)=Pr(X -1 (x)) Probability density function f(x), cumulative distribution function F(x)=Pr(X x), reliability function (complementary distr. Function) R(x)=1-F(x)=Pr(X>x) Expected value, moments: E(X n )= x n f(x) dx Relevant Examples, e.g.: number of packets that arrive at the access router in the next hour (discrete) Buffer occupancy (#packets) in switch x at time y (discrete) Number of downloads ( mouse clicks ) in the next web session (discrete) Time until arrival of the next IP packet at a base station (continuous) Page 5 Basic concepts: Exponential Distributions Important Case: Exponentially distributed RV Single parameter: rate Density function f(x)= exp(- x), x>0 Cdf: F(x)=1-exp(- x), Reliability function: R(x)=exp(- x) Moments: E{X}=1/ ; Var{X}=1/ 2, C 2 = Var{X} / [E{X}] 2 = 1 Important properties: Memory-less: Pr(X>x+y X>x) = exp(- y) Properties of two independent exponential RV: X with rate, Y with rate µ Distribution of min(x,y): exponential with rate (+µ) Pr(X<Y)= /(+µ) Page 6

4 Basic concepts III: Stochastic Processes Definition of process (X i ) (discrete) or (N t ) (continuous) Simplest type: X i independent and identically distributed (iid) Relevant Examples: Inter-arrival time process: X i Counting Process: n-1 N(t) = max{n Σ i=1 X i t}, alternatively N i ( ) = N(i ) N([i-1] ) Important Example: Poisson Process Assume i.i.d. exponential packet inter-arrival times (rate ): X i :=T i -T i-1 Counting Process: Number of packets N t until time t Pr(N t =n)= (t) n exp(- t) / n! Properties: Merging: arrivals from two independent Poisson processes with rate 1 and 2 Poisson process with rate ( ) Thinning: arrivals from a Poisson process of rate are discarded independently with probability p Poisson process with rate (1-p) Central Limit Theorem: superposition of n independent processes results in the limit n in a Poisson process (under some conditions on the processes) Page 7 Content 1. Motivation & Background Performance Analysis in Wireless Settings Review of Basic Concepts: Random Variables, Exponential Distributions, Stochastic Processes 2. Simulation Models Basics: Discrete Event Simulation Random Number Generation Output Analysis 3. Simple Analytic Models Birth-Death Processes M/M/1 Queues Circuit-Switched case: Erlang formula Packet-based traffic models 4. Summary Page 8

5 Simulation Models (I) Basic principles of discrete event simulation Virtual simulation time t System state S(t) Events occur at certain times t i Instantaneous changes of system state S(t i-1 ) S(t i ) Possibly scheduling of follow-up events Events stored in ordered event list System description: Entities, attributes, and activities Frequently object oriented implementation Important aspects Initial state S(0) Termination Criterion Fixed simulation time T Fixed number of packets/connections Occurrence of certain events (e.g. Loss of connectivity) Page 9 Simulation Models (II) Application to wireless networks:main components Topology definition: nodes and connectivity Link properties: e.g., Propagation models Node functionalities: e.g., schedulers, buffer management, L2/L3 protocol implementation Traffic models (and transport protocol implementation) Mobility Models probabilistic elements in several of these components stochastic simulation alternative : trace-driven simulations Output parameters, statistics collection, e.g. Packet based End-to-end packet delay packet loss rate energy per packet Connection based File Transfer times Fraction of blocked calls throughput Node/Link Properties Buffer occupancy Link utilizations throughput Page 10

6 Types and Examples of Simulation Tools Libraries and programming languages with basic functionalities and data types: Sim_lib [Watkins, Kevin: Discrete Event Simulation in C, 1993] Simula [e.g. R. Pooley: An Introduction to Programming in SIMULA, 1987] General Purpose Simulation Environments, e.g. DEMOS/MAOS [Birtwistle, A system for discrete event modelling on SIMULA (DEMOS), 1979] GPSS [ Network Simulation Tools, e.g. NS2 [ OPNET WIPSIM [ Glomosim [ Page 11 Random Number Generation Uniform Random Number Generator (RNG) Sequence U 1,U 2,... of i.i.d. random numbers, uniformly distributed in ]0,1[ Pseudo-random: same seed X 0 same sequence example: linear congruential generator X i+1 =(a X i + b) mod c, U i+1 =X i+1 /c E.g. a=7 5 =16807, b=0, c= (prime) Random Variables from general distributions Y 1,Y 2, with cumulative distribution function F(x) derived from uniform stream U 1,U 2, by Inversion: Y i =F -1 (U i ) Other Techniques: Rejection, Convolution/Composition, etc. Page 12

7 Output Analysis (I): General Goal: Obtain Estimator Ž of desired performance parameter µ Note: Ž often multi-dimensional Ž is a random variable with some distribution f Ž (x) Considered case here: Ž is estimator of µ=e(z) Properties of estimator: Ž t is called (t=simulation time) Unbiased when E(Ž t )=µ Consistent when lim Ž t =µ for t (stochastic convergence) Types of Simulations Terminating simulations ~ transient parameters Non-terminating simulations ~ Steady-state parameters Page 13 Output Analysis II: Terminating Simulations Terminating Simulations Explicite stopping criterion, e.g. Fixed simulation time Fixed number of arrivals/connections Specific event (e.g. Buffer overflow, component failure) Approach: Independent Replications Repeat Experiment R times, each time with different seeds independent outcomes Z 1,Z 2,...Z R Estimator Ž=1/R Σ Z i Unbiased Asymptotically normal distributed Relevant examples: Determine average buffer-occupancy during busy hours 9-17hrs (starting empty at 9hrs) Determine probability that call will be dropped before its desired end (given initial conditions) Determine probability of buffer-overflow within n packet arrivals (given empty buffer in beginning) Page 14

8 Output Analysis III: Confidence Intervals Example: Estimate Probability γ=pr(overflow before simulation time t) R replications with indep. outcomes Z i Estimator Ž=1/R Σ Z i E(Ž)= γ (unbiased!), Estimates Š 2 of Var(Z i ) Š 2 = 1/(R-1) Σ (Z i -Ž) 2 Š 2 /R is estimate of σ 2= Var(Ž) Approaches for Confidence Intervals, confidence level 1-α (often 1-α =95%): Convergence to normal distribution (Ž- γ)/ σ in the limit standard normal distributed Hence for n 1-α/2 =quantile of normal distribution at level (1-α)/2 Pr(Ž- σ n 1-α/2 < γ < Ž + σ n 1-α/2 ) = 1-α Using the variance estimate Š 2 /R: Pr(Ž- Š 2 / R n 1-α/2 < γ < Ž + Š 2 / R n 1-α/2 ) = 1-α General variance estimate use of Student t distribution (but only exact when Z i normal!) R (Ž- γ)/š approx. Student-t distribured with (R-1) degrees of freedom Other approaches, e.g. variance stabilization for probability estimates [see Heyman/Sobel] Page 15 Special case: Binary Outcome Example: Estimate Probability γ=pr(overflow before simulation time t) R replications with indep. outcomes Z i ={1 when overflow occurred, 0 otherwise} Estimator Ž=1/R Σ Z i R* Ž Binomially distributed: expected value γr, variance R γ(1- γ) E(Ž)= γ (unbiased!), Var(Ž)=γ(1- γ)/r Estimates Š 2 of Var(Z i )=σ 2 Š 2 = Ž(1-Ž) [for probabilities] Page 16

9 Output Analysis IV: non-terminating case Steady state parameters in theory infinite simulation needed Finite simulation length causes biased estimator Approaches: Independent replications, impact of transient phase avoid transient phase Single, long simulation run Problem: correlated samples require adjustment of variance estimate Alternatives Batching Regenerative Simulation Page 17 Content 1. Motivation & Background Performance Analysis in Wireless Settings Review of Basic Concepts: Random Variables, Exponential Distributions, Stochastic Processes 2. Simulation Models Basics: Discrete Event Simulation Random Number Generation Output Analysis 3. Simple Analytic Models Birth-Death Processes M/M/1 Queues Circuit-Switched case: Erlang formula Packet-based traffic models 4. Summary Page 18

10 Continuous Time Markov Processes Defined by State-Space: finite or countable infinite, w/o.l.g. E={0,1,2,...,K} (K= also allowed) Transition rates: µ jk Holding time in state j: exponential with rate Σ k j µ jk Transition probability from state j to k: µ jk / Σ l j µ jl =: µ jk / µ j X t = RV indicating the current state at time t; π i (t):=pr(x t =i) Markov Property : transitions do not depend on history but only on current state t 0 <t 1 <...<t n, i 0,i 1,...i n,j E Computation of steady-state probabilities Chapman Kolmogorov Equations: dπ i (t)/dt = - µ i π i (t) + Σ j i µ ji π j (t) Flow-balance equations, steady-state: µ i π i (t) = Σ j i µ ji π j (t) Here: restriction to irreducible, homogeneous processes Page 19 Queueing Models: Kendall Notation X/Y/C[/B] Queues (example: M/M/1, GI/M/2/10, M/M/10/10,...) X: Specifies Arrival Process M=Markovian Poisson GI=General Independent iid Y: Specifies Service Process (M,G(I),...) C: Number of Servers B: size of finite waiting room (buffer) [also counting the packet in service] If not specified: B= Often also specified: service discipline FIFO: First-In-First-Out (default) Processor Sharing: PS Last-in-first-out LIFO (preemptive or non-preemptive) Earliest Deadline First (EDF), etc. µ Finite buffer (size B) Scope here: Point-process models as opposed to fluidflow queues Page 20

11 M/M/1 queue Poisson arrival of packets (first M Markovian) with rate Exponentially distributed service times of rate µ (second M ) Single Server (1) FIFO service discipline Q t = Number of packets in system is continuous-time Markov Process Derived Parameter: Utilization, ρ= / µ : if ρ 1, instable case (no steady-state q.l.d) Infinite buffer µ Performance Parameters Queue-length distribution: π(t), steady-state limit: π=lim π(t) (if ρ<1) Queue-length that an arriving customer sees Waiting/System time distribution Buffer-Overflow Probability for level B = Pr(arriving customers sees buffer occupancy B or higher) Page 21 M/M/1 queue: Performance µ µ µ Birth-Death Process Probability of i packets in queue [using flow-balance equations] π i := Pr(Q=i) = (1-ρ)* ρ i, where ρ= / µ <1 Probability of idle server: π 0 = (1-ρ) Average Queue-length: E{Q}= ρ/(1-ρ) Average Delay (System Time): E{S}= E{Q}/ = 1/(µ-) Buffer Overflow Probabilities (PASTA principle) Pr(Q (a) B)= Pr(Q B) = ρ B Page 22

12 General Birth-Death Processes µ 1 µ 2 µ 3 Steady-State Probabilities (from balance equations): π i := Pr(Q=i) = π 0 k=0 i-1 k / k=1 i µ k Models in this class, e.g. M/M/1/B M/M/C, M/M/C/C Load-dependent services, discouraged arrivals Page 23 The circuit switched scenario K channels Users allocate one channel per call for certain call duration If all channels are allocated additional starting calls are blocked How many channels are necessary to achieve a call certain maximal blocking probability? Common Model Assumptions: Calls are arriving according to a Poission Process (justified for large user population, limit theorems for stochastic processes) with rate Call durations are exponentially distributed with mean T (okay for voice calls) Page 24

13 Computation of blocking probabilities: M/M/K/K model, Erlang-B formula K 1/T 2/T 3/T K/T Finite Birth-Death Process: Probability of i calls active π i := Pr(n=i) = π 0 (T) i /i!, i=1,,k where π 0 = 1/[Σ(T) i /i!] (sum taken over i=0 to K) Probability of blocked call: p (Blocking) = π K = π 0 (T) K /K! [also known as Erlang-B formula] Page 25 Packet-based link model: M/M/1/K queue Assumptions Poisson arrival of packets with rate Exponentially distributed service times of rate µ Single Server Finite waiting room (buffer) for K packets Suitable e.g. for modeling bottleneck link in packet-based wireless networks [Full network models: see traffic analysis lecture] Finite buffer (size K) µ Page 26

14 M/M/1/K queue: Performance K µ µ µ µ From Birth-Death Process Theory: Probability of i packets in queue π i := Pr(Q=i) = (1-ρ)/(1- ρ K+1 ) * ρ i, where ρ= / µ 1, i=0,,k Probability of packet loss: p (loss) = π K = (1-ρ)/(1- ρ K+1 ) * ρ K Average Delay: Ď = 1/[ (1-p K )] * ρ/(1- ρ K+1 ) * [(1- ρ K )/ (1-ρ) K ρ K ] Page 27 Extension: Models for packet traffic Poisson assumption for packet arrivals may be applicable for highly aggregated traffic (core networks), but otherwise traffic tends to be bursty High data rates in ftp download but less activity between downloads http: activities after mouse-clicks Video streaming: high data rates in frame transmissions Interactive Voice: talk and silent periods Model Modifications: Bulk Arrival processes ON/OFF models Hierarchical models Page 28

15 Bulk Arrival Models Queue-length at arrival instances increases not only by 1, but by a Random Variable B, the bulk-size Parameter set of model Bulk arrival process, e.g. exponential with rate Bulk-Size distribution: p i (e.g. geometric) Service rate (single packet) Steady-state solution for mean system time [Chaudhry & Templeton 83]: E{S} = [ E{B}+E{B 2 } ] / [2 E{B} µ (1-ρ) ] Example: M (B) /M/1 queue with geometrically distributed B Page 29 More realistic models: ON/OFF Models Parameters: N sources, each average rate κ During ON periods: peak-rate p bursty traffic, when p >> κ Mean duration of ON and OFF times κ = p ON/(ON+OFF) Page 30

16 Traffic models: General hierarchical models Frequently used: Several levels with increasing granularity E.g. 3 levels: sessions, connections, packets Or: 5-level model: Page 31 Example: HTTP traffic model Main objects contain zero or more embedded objects that the browser retrieves Correlated requests for embedded objects within retrieval of main object start browser HTTP Session (User A) HTTP Session (User B) click HTTP Session (User C)... click Download Phase 1 Idle time Download Phase 2 Dld. Phase 3... Read time click Dld. Phase K exit browser Get Main Object Get embedded Obj. 1 Get emb. Obj Get emb. Obj. N Session Level Connection/ Flow Level Packet Level, TCP dynamics (not shown here) Statistics: Session arrivals: Renewal process (Poisson) Idle time: heavy-tail # embedded objects: geometric (measurements e.g. mean 5) Object size: heavy-tailed Page 32

17 Summary 1. Motivation & Background Performance Analysis in Wireless Settings Review of Basic Concepts: Random Variables, Exponential Distributions, Stochastic Processes 2. Simulation Models Basics: Discrete Event Simulation Random Number Generation Output Analysis 3. Simple Analytic Models Birth-Death Processes M/M/1 Queues Circuit-Switched case: Erlang formula Packet-based traffic models 4. Summary Page 33 Exercises: 1. Simulations: Use the simulation program from MM1 to obtain blocking probabilities for GPRS sessions. Compute the 95% confidence intervals for these blocking probabilities via a) A single simulation run, using binomial distributions b) Independent replications and the normal approximation for the average blocking probability. 2. Analytic Models: Traffic measurements in a GPRS radio cell result in the following traffic model: voice calls arrive at Poisson rate 1call/min and have an average duration of 1.5 min. GPRS data sessions start at rate 1session/5min, have an average duration of 20min, and generate traffic with an averate rate of 10kb/sec using IP packets of 1500 byte size and CS-II. a) How many time-slots would have to be reserved to GSM voice calls to keep the call blocking probability below 1e-6? b) Compute the average RLC frame delay, if 4 GPRS time-slots are used for the data traffic (as simplification: use an M/M/1 queue on RLC layer, neglecting header overhead as well as the overhead of TBF assignments). [c) (optional): Develop a Markov model for a queueing model for packet switched traffic under the assumption that 6 time-slots are available in total (shared by voice and date) when voice traffic has preemptive priority over packet traffic. Compute the average queue-length for the RLC layer queue. ] Page 34

18 References Analytic Models Cassandras, Lafortune, Introduction to Discrete Event Systems, Chapts. 7 and 8, Kluwer, more details in lecture Traffic analysis I (H. Schiøler, 8th Sem DIRS/NPM) Simulation models Cassandras, Lafortune, Introduction to Discrete Event Systems, Chapt. 10, Kluwer, Heyman, Sobel (ed.), Stochastic Models, Chapt. 6 and 7, North-Holland, more details in lecture Discrete Event simulation (9th Sem DIRS/NPM) Page 35