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**Discrete Event Simulation**

Speaker: Yeh, En-Hau Advisor: Phone Lin MCN Lab., Dept. of CSIE, NTU TEL: #538

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**Grading Midterm Exam 40% Final Exam 30% Final Project 30%**

Step 1: Find a paper with the analytical model Step 2: Read the paper in detail Step 3: Write the simulation to validate the analytical model Step 4: Compare the errors of your simulation results with the results in the paper [optional] Step 5: Do some more if you can (e.g., performance analysis via simulation or analytical models). Step 6: Prepare your presentation Report

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**Grading Find a paper with the analytical model Where to find a paper?**

IEEE Xplore: ACM Digital Library: Which Journal/Magazine? Journals IEEE Transactions on Vehicular Technology IEEE Journal on Selected Areas in Communications IEEE Transactions on Mobile Computing IEEE Networks IEEE Transactions on Computer IEEE Communications Letter IEEE Transactions on Wireless Communication IEICE Transactions on Communication Security and Communication Networks ACM/Springer Wireless Networks Journal of Wireless Communications and Mobile Computing ACM/Springer Mobile Networks and Applications Magazines IEEE Personal Communications Magazine IEEE Communications Magazine IEEE Wireless Communications Magazine

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**Outline Introduction to Simulation Simulation Implementation**

What is Simulation? Why Simulate? System & Simulation Simulation Implementation A Simulation Example M/M/c/c Handoff Model

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What is Simulation? Simulation is the imitation of some real thing, state of affairs, or process. Simulation is defined as the process of creating a model of an existing or proposed system to gain some understanding of how the corresponding system behaves.

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What is Simulation?

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**Why Simulate? (Example)**

stack landing

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**Why Simulate? (Example Cont.)**

An Example of Simulation in the Real World The Airport Problem The airport has two runways, one for landing and one for takeoff. If the landing way is busy, the airplane in the air must enter in a stack. However, when the stack is full, the takeoff way must become landing way. Airports have a waiting stack, where airplanes wait before they can land. In this stack, airplanes ﬂy a holding pattern on an assigned altitude. What we want to know? Calculate the stack length. The average time a plane waits for takeoff. The average utilization of the landing way and the takeoff way.

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**Why Simulate? (Example Cont.)**

Such measures of performance can get from historical data records. However, we are interested in the impact of certain proposed changes. Landing airplane arrival rate. Takeoff airplane arrival rate. Increasing the runways.

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**Why Simulate? Save time Reduce cost**

Test design ideas (performance measurement) The impact of certain changes of parameter. The arrival rate of certain traffic class. The service time of certain traffic class. The number of application server. Predict the future behavior of the system

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**Simulation Static/Dynamic Simulation Continuous/Discrete Simulation**

Time simply plays no role Dynamic A system evolves over time Continuous/Discrete Simulation Continuous The state change continuously with respect to time Discrete The state change instantaneously at separated points in time

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**Discrete-Event Simulation**

Dynamic Simulation clock Keep track of the current value of simulated time as the simulation proceeds A mechanism to advance simulated time from one value to another Discrete The state change instantaneously at separate points in time The system can change at only a countable number of points in time. These points in time are the ones at which an event occurs. Event An instantaneous occurrence that may change the state of the system

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**Discrete-Event Simulation**

All state changes occur only at event times for a discrete-event simulation model Periods of inactivity are skipped over by jumping the clock from event time to event time Stochastic Interarrival times A1, A2,… and service times S1, S2,… Are random variables Have cumulative distribution functions For example, the process of arrival at and departure from a voip server occurs in random fashion. The only way we can represent this randomness is by using probability distributions to describe.

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**The Phase of Simulation**

Construction Formularize the problem Collecting necessary input data as parameter (arrival rate, departure rate) Design a simulation model to match the problem Running Run this simulation for enough time to estimate the system’s result. Debugging See if the result make sense. Compare to an analytic model

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**Simulation Implementation**

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**Simulation Implementation**

Event An instantaneous occurrence that may change the state of the system Attributes of an event Type (e.g., arrival, departure, etc.) Timestamp Other attributes Location info. Programming Language C C++ Java C# … Priority Queue in C++ Standard Template Library priority_queue<T, Sequence, Compare> Event List Event list is a linked list Events are sorted according to the timestamp value. Automatically adjusts the event order at each event insertion. Efficiency Data structure: link-list v.s. min-Heap data structure Time Complexity Memory allocation management Freelist

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**Random Number Generator**

R.N.G.’s for various probability distributions Used when we need a time period with a certain distribution Uniform, exponential, gamma, etc. Implementation The inverse transform method CDF: F(x) = Pr[ X ≤ x]

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**Inverse Transform Method**

1 F(x) yj F-1(yj) yi Probability Uniform(0,1) xi xj X

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**Inverse Transform Method**

Validity of this method: Whether r.v. F-1(U) has the CDF F(x)? F’(x) = Pr[F-1(U) ≤ x] = Pr[U ≤ F(x)] = FU( F(x) ) = F(x)

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**Inverse Transform Method**

Example: exponential distribution with rate CDF: F(x) = 1 - e -λX Set F(X) = U => 1 - e-λX = U => e-λX = 1-U => -λX = ln(1-U) => X = -ln(1-U)/ λ Note: u!=0

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**Inverse Transform Method**

Applicable when the inverse function of F(x), i.e., F-1(x), can be derived Preferably when F-1(x) has an east-to-use form Steps: Generate a uniform (0,1) random number u Compute x = F-1(u), and x is the desired random number

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**Generating Gamma-distributed**

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**Generating Gamma-distributed**

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**A SIMULATION EXAMPLE M/M/c/C**

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**Queueing System Key elements: customers and servers**

Customer: arrival process Server: service rate and number of servers Notation: A/B/c/N/K A: inter-arrival time distribution (arrival process) B: service time distribution c : number of servers/channels N: system capacity K: queue discipline (Unless specified, it is assumed to be First-In-First-Out)

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**Queueing System M/M/c/c The 1st M: Markovian / Memoryless**

Possision arrivals / exponential inter-arrival time The 2nd M: Markovian Exponential service time c: c servers/channels c: system capacity Blocking

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**Analysis: M/M/c/c (Erlang B)**

λ λ λ λ λ … 1 2 c-1 c μ 2μ 3μ (c-1)μ cμ

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**Simulation Models: M/M/c/c**

ARRIVAL event represents a customer arrival. arrival1 arrival2 departure1 generate next call: arrival2 if channel > 0 then channel-- and generate service time: departure1

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**Simulation Models: M/M/c/c**

arrival1 arrival2 arrival3 departure1 departure2 generate next call: arrival3 if channel > 0 then channel-- and generate service time: departure2

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**Simulation Models: M/M/c/c**

DEPARTURE event represents a customer departure . arrival1 arrival2 arrival3 departure1 departure2 channel++

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**Simulation Models: M/M/c/c**

Assume that c=2 Arrival5 is Blocking Pb = N_blocking / N_arrival arrival1 arrival2 arrival3 arrival4 arrival5 arrival6 departure1 departure2 departure4 generate next call: arrival6 if channel = 0 then N_blocking++

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**Simulation Models: Event Class**

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**Simulation Models: Initialize**

實驗次數100萬次

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**Simulation Models: Timing Routine**

generate next call if channel > 0 then channel generate service time if channel = 0 then N_blocking++ channel++

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**Simulation Models: Report**

M/M/c/c Inter-arrival time is Exponential distribution with rate 50.0 Service time is Exponential distribution with rate 10.0 c: number of channels is 8 c: system capacity is 8

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Priority Queue

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**A SIMULATION EXAMPLE Handoff Model**

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**A Simulation Example Handoff Model**

When the network attempts to deliver a call to a handset or the handset attempts to originate a call, the call is connected if a channel is available. Otherwise, the call is blocked (referred to as the new call blocking). When a handset moves from one cell to another during a call, in order to maintain call continuation the channel in the old cell is released, and a channel is required in the new cell. This process is called hand-off. If no channel is available in the new base station, then the hand-off call is force-terminated.

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**A Simulation Example Handoff Model**

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**System Model – Assumptions (1/4)**

Each cell consists of One base station Several channels Several mobile station A mobile user can carry only a call We do not consider batch arrivals of new and handoff calls

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**System Model – Assumptions (2/4)**

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**System Model – Assumptions (3/4)**

Each hard handoff call can only connect to one BS.

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**System Model – Assumptions (4/4)**

Wrapped mesh cell: N*N => 8*8 The boundary cell effects can be ignored

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**Simulation Models: Event Design**

A HANDOFF event An ARRIVAL event A RELEASE event

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**Simulation Models: Event Design**

We define three types of events for traditional calls : ARRIVAL event represents a new call arrival. Call_Type is maintained in this event to indicate that the event is for a (Call_type = Traditional) A channel is required HANDOFF event represents a handoff call arrival. The Call_Type variable is used to indicate that the event is for a traditional call (Call_type = Traditional) The channel in the old cell is released, and a channel is required in the new cell. RELEASE event represents the completion of a call. Call_Type indicates that the event is for a traditional call (Call_type = Traditional) The channel is released.

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**Timing Diagram for one call**

tc: the call holding time of a portable. tc,i: the period between the time the portable moves into coverage area i and the time when the call is complete. tm,i: the residence time of a portable at a coverage area i. τ0: the time between the arrival of the call and when the portable moves out of coverage area 0.

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Residual-life τ0 Generation of an residual-life random number for gamma residence time with the parameters (k, θ) includes the following steps: We first generate a uniform random number μ in (0, 1). Second, we generate a random number t for the gamma random variable T with the parameters (k+1, θ). By multiplying u and t, we obtain a random number for the residual τ0. τ0=U*Gamma(k+1, θ) Yi-Bing Lin, “Random Number Generation for Residual Life of Mobile Phone Movement”, IEEE International Conference on Networking, Sensing and Control, Vol 1, pp , March 2004.

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**Timing Diagram for multiple call**

first call Event List Call 1 Arrival (t=0) Call 1 Handoff (t=4) Call 2 Arrival (6) Call 2 Arrival (t=6) generate next call arrival2 generate service time generate cell residence time insert ARRIVAL event if channel>0 { channel-- if (service time > residence time) insert HANDOFF event else insert RELEASE event }

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**Timing Diagram for multiple call**

Event List Call 1 Arrival (t=0) Call 1 Handoff (t=4) Call 2 Arrival (6) Call 2 Arrival (t=6) Call 1 RELEASE (t=14) Call 3 Arrival (t=20) channel++ generate next call arrival3 generate cell residence time if channel==0 N_Blocking++ compare the service time and the residence time if (service time > residence time) insert HANDOFF event else insert RELEASE event

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**Example: Event List and Event Struct **

…

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**Simulation Models: Timing Routine**

/* initialization */ /* generate the first event and insert it into the event list */ While(Number_of_Case <= Bound){ /* retrieve the first event from event list */ /* adjust the system clock according to the timestamp of current event */ switch(event_type){ case ARRIVAL : /*Process flowchart for ARRIVAL event */ break; case HANDOFF: /*Process flowchart for HANDOFF event */ case RELEASE: /*Process flowchart for RELEASE event */ } /*Calculate the desired performance measures and output the result */

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Flow Chart

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The notations The new traditional call arrival rate to a cell is exponentially distributed with the rate The call holding time for a traditional call is exponentially distributed with the mean The UE residence time in a cell has an exponential distribution with the mean The total channels in a cell are C. The call incompletion probability for a traditional call is Pnc,t.

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**Analysis and Simulation Validation**

The input parameters and are normalized by . We assume there are C=10 channels per cell.

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**Reference http://www.erlang.com/calculator/erlb Gamma distribution**

Erlang B Calculator Latex GNUPlot M/M/c/c Sample Source Code

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