Random Sum IID RVs. N is a random variable

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1 Answer. # 61

2 Outline Introduction to Poisson Processes Definition of arrival process Definition of renewal process Definition of Poisson process Properties of Poisson processes Inter-arrival time distribution Waiting time (Arrival Time) distribution Superposition and decomposition Non-homogeneous Poisson processes (relaxing stationary) Compound Poisson processes (relaxing single arrival) Modulated Poisson processes (relaxing independent) 62

3 Random Sum IID RVs N is a random variable 63

4 Compound Poisson Processes (Relaxing Single Arrival) A stochastic process process if it can be represented as is said to be a compound Poisson is a Poisson process with mean λ is a family of i.i.d. random variables that is independent of 64

5 Proof 65

6 # # 66

7 Example Solution 67 #

8 Example (Batch Arrival Process) 68

9 Solution 69

10 ln Poisson moment generating function Geometric s moment generating function # 70

11 Modulated Poisson Processes (Relaxing Independent Increment) Assume that there are two states, 0 and 1, for a "modulating process." arrival rate λ arrival rate λ 1 When the state of the modulating process equals 0 then the arrive rate of customers is given by λ0, and when it equals 1 then the arrival rate is λ1. The residence time in a particular modulating state is exponentially distributed with parameter μ and, after expiration of this time, the modulating process changes state. The initial state of the modulating process is randomly selected and is equally likely to be state 0 or 1. 73

12 Modulated Poisson Processes (con t) For a given period of time (0, t ), let be a random variable that indicates the total amount of time that the modulating process has been in state 0. Let X(t) be the number of arrivals in (0, t ). Then, given, the value of X(t) is distributed as a non-homogeneous Poisson process and thus The difficulty in determining the distribution for X(t) is to calculate the density of. There are some limiting cases that are of interest. As μ 0, the probability that the modulating process makes no transitions within t seconds converges to 1, and we expect for this case that 74

13 Modulated Poisson Processes (con t) As μ, then the modulating process makes an infinite number of transitions within t seconds, and we expect the modulating process to spend an equal amount of time in each state such that Example (Modeling Voice). A basic feature of speech is that it comprises an alternation of silent periods and non-silent periods. The arrival rate of packets during a talk spurt period is Poisson with rate λ 1 and silent periods produce a Poisson rate with λ 0 0. The duration of times for talk and silent periods are exponentially distributed with parameters, respectively. The model of the arrival stream of packets is given by a modulated Poisson process. 75

14 Interrupted Poisson Process (IPP) Poisson process with rate ON OFF Stay in ON state for a period exponentially distributed with mean 1/ Stay in OFF state for a period exponentially distributed with mean 1/

15 Markov Modulated Poisson Process (MMPP) Example: 3-state MMPP Poisson process with rate 2 p 12 Poisson process with rate 1 p Stay in state i for a period exponentially distributed with mean 1/ i p 13 p 32 p 23 p 31 3 Poisson process with rate 3

16 Memoryless Property of the Exponential Distribution A random variable X is said to be memoryless or without memory, if (3) The condition in Equation (3) is equivalent to or (4) Since Equation (4) is satisfied when X is exponentially distributed (for ), it follows that exponential random variable are memoryless. Not only is the exponential distribution " memoryless," but it is the unique continuous distribution possessing this property. 78

17 Example 79

18 Solution # 80

19 Example 81

20 Solution # exponential pdf: λ e λx cdf: 1 e λx mean: λ 1 82

21 Minimum of Exponential Random Variables # 83

22 Comparison of Two Exponential Random Variables exponential # pdf: λ e λx cdf: 1 e λx mean: λ 1 84

23 Example 85

24 Solution Minimum of R 1 and R 2 86

25 87 #

26 Maximum of Exponential Random Variables 88

27 Example 89

28 Solution 90

29 Solution 91

30 Example 92

31 Solution 93

32 Example 94

33 Solution 95

34 Example 96

35 Solution 98

36 99 #

37 100 #

38 Solution II for (c) 101

39 Example Solution 102

40 Solution 1. #

41 #

42 Queuing System Queue Queuing Time Server Service Time Response Time (or Delay) 105

43 Queuing Theory for Studying Networks View network as collections of queues FIFO data-structures Queuing theory provides probabilistic analysis of these queues Examples: Average length Average waiting time Probability queue is at a certain length Probability a packet will be lost 106

44 Little s Law The long-term average number of customers in a stable system N, is equal to the long-term average arrival rate, λ, multiplied by the long-term average time a customer spends in the system, T. Expected number of customers in the system E N E T Expected time in the system Arrival rate IN the system 107

45 Generality of Little s Law Mean number tasks in system = mean arrival rate x mean response time E N E T Little s Law is a pretty general result It does not depend on the arrival process distribution It does not depend on the service process distribution It does not depend on the number of servers and buffers in the system. Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks λ Aggregate Arrival rate Queueing Network

46 Characteristics of queuing systems Arrival Process The distribution that determines how the tasks arrives in the system. Service Process The distribution that determines the task processing time Number of Servers Total number of servers available to process the tasks 109

47 Specification of Queueing Systems Arrival/Departure Customer arrival and service stochastic models Structural Parameters Number of servers: What is the number of servers? Storage capacity: are buffer finite or infinite? Operating policies Customer class differentiation are all customers treated the same or do some have priority over others? Scheduling/Queueing policies which customer is served next Admission policies which/when customers are admitted

48 Kendall Notation A/B/m(/K/N/X) To specify a queue, we use the Kendall Notation. The first three parameters are typically used, unless specified A: Inter arrival distribution B: Service time distribution m: Number of servers (1, 2, ) K: Storage Capacity (1, 2,, infinite if not specified) N: Population Size (1, 2,, infinite if not specified) X: Service Discipline (FCFS/FIFO/RSS) 111

49 Kendall Notation of Queueing System Arrival Process M: Markovian D: Deterministic Er: Erlang G: General A/B/m/K/N/X Service Process M: Markovian D: Deterministic Er: Erlang G: General Number of servers m=1,2, Service Discipline FIFO, LIFO, Round Robin, Storage Capacity K= 1,2, (if then it is omitted) Number of customers N= 1,2, (for closed networks, otherwise it is omitted) 112

50 Distributions M: stands for "Markovian", implying exponential distribution for service times or inter-arrival times. D: Deterministic (e.g. fixed constant) E k : Erlang with parameter k H k : Hyper-exponential with parameter k G: General (anything) 113 CS352 Fall,2005

51 Kendall Notation Examples M/M/1 Queue Poisson arrivals (exponential inter-arrival), and exponential service, 1 server, infinite capacity and population, FCFS (FIFO) the simplest realistic queue M/M/m Queue Same, but m servers M/D/1 Queue Poisson arrivals and CONSTANT service times, 1 server, infinite capacity and population, FIFO. G/G/3/20/1500/SPF General arrival and service distributions, 3 servers, 17 queues (20-3), 1500 total jobs, Shortest Packet First 114

52 Performance Measures of Interest We are interested in steady state behavior Even though it is possible to pursue transient results, it is a significantly more difficult task. E[S]: average system (response) time (average time spent in the system) E[W]: average waiting time (average time spent waiting in queue(s)) E[X]: average queue length E[U]: average utilization (fraction of time that the resources are being used) E[R]: average throughput (rate that customers leave the system) E[L]: average customer loss (rate that customers are lost or probability that a customer is lost)

53 Queue A memoryless, Poisson process always gives an exponential distribution for inter-arrival or inter-service times However there are cases where the service process times are not exponentially distributed Ex.1 A packet switching system may handle only two different types of packet, one with 100 bytes, and one with 2,000 bytes. The big packets will take longer to serve (transmit) F(t) A dumbbell distribution: The service times of the small packets would cluster around a low value, and there would be a cluster at a longer time for the larger packets Service time, t 116

54 Queue (con t) Ex.2 A server (computer) has to perform different operations on packets, depending on what type of packet it is Possible operations are encrypting, processing for an on-line game, and simple transmission Ex.3 Within an ATM switch, where all packets (or cells ) are the same size, and thus take the same time to transmit In these cases, the service time distribution is said to be general, and we describe the queue as M/G/1 Customer arrival: Poisson with rate λ Service times: i.i.d. general distribution G, independent to the arrival distribution 117

55 Busy Period of a Queue S k : The time until k additional customers have arrived S k has Erlang distribution with parameters (k, λ) Y 1, Y 2, : The sequence of service times The busy period will last a time t and consist of n services iff S k Y 1 +Y 2 + Y k, k = 1, n-1 Y 1 +Y 2 + Y n = t There are n-1 arrivals in (0, t) S 1 S 2 S k (total n customers are served) 0 Y 1 Y 2 Y k 120

56 idle period S 1 S 2 S n-1 S n 0 t Y 1 Y 2 Y n-1 Y n Busy Period (1) (2) 121

57 (1) 122

58 0 t Y 1 Y 2 Y n-1 Y n Y 1, Y 2, Y n are of i.i.d. general distribution G 123

59 (2) The arrival process is independent of the service times G n : n-fold convolution of G with itself convolution of x() an h() 124

60 125 #