Note10 - Computer Networks Modeling arrivals and service with Poisson Saad Mneimneh Computer Science Hunter College of CUNY New York Now remember

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Unformatted text preview: Computer Networks Modeling arrivals and service with Poisson Saad Mneimneh Computer Science Hunter College of CUNY New York Now remember this: A poisson has a short memory span of few months, but a Poisson process is memoryless. 1 Introduction In computer networks, packet arrivals and service are modeled as a stochastic process in which events occur at times t 1 ,t 2 ,... For instance, in the figure below, t 1 ,t 2 ,... can be interpreted as the packet arrival times, or the service completion times. Accordingly, T i , defined as t i- t i- 1 for i > 1, can be interpreted as the inter-arrival times of packets (intervals between subsequent arrivals), or the delays experienced by the served packets (assuming that the server is always busy). Similarly, A ( t ) denotes the number of packets that arrive in [0 ,t ], or the number of packets served in [0 ,t ]. In the rest of this document, we will refer to packets rather than service, but it should be clear that the discussion applies to both. t 1 T 1 T 2 T 3 A ( t ) t 2 t 3 Figure 1: Arrival/service process 1 2 Poisson The packet inter-arrival times are typically modeled as a Poisson process, i.e. T i are independent and identically distributed (IID, so we drop i from the term T i in the following expression), and obey an an exponential distribution: F T ( t ) = P ( T ≤ t ) = 1- e- λt where λ is a parameter. We will give λ a name shortly. The probability density function for T is therefore (the derivative of 1- e- λt with respect to t ): f T ( t ) = λe- λt Therefore, P ( t 1 ≤ T ≤ t 2 ) = Z t 2 t 1 λe- λt =- e- λt | t 2 t 1 =- e- λt 2 + e- λt 1 The expected value of T can be obtained as: E [ T ] = Z t tλe- λt dt = 1 λ Therefore, the parameter λ is called the arrival rate, or simply rate 1 . Similarly, E [ T 2 ] = Z t t 2 λe- λt dt = 2 λ 2 Therefore, the variance is: σ 2 ( T ) = E [( T- E [ T ]) 2 ] = E [ T 2 + E [ T ] 2- 2 E [ T ] T ] By the linearity of expectation, σ 2 ( T ) = E [ T 2 ] + E [ T ] 2- 2 E [ T ] E [ T ] = E [ T 2 ]- E [ T ] 2 = 1 λ 2 3 Poisson process is memoryless Now we prove a unique property of the exponential process, known as the memoryless property. Consider the waiting time until some arrival occurs. The memoryless prop- erty states that given that no arrival has occurred by time τ , the distribution of the remaining waiting time is the same as it was originally. Mathematically, P ( T > τ + t | T > τ ) = P ( T > t ) The proof is simple as a direct consequence of the exponential distribution: 1 A mathematical result, known as the law of large numbers, says that if X i are IID, then lim n →∞ ∑ n i =1 X i n = E [ X ]. Therefore, the inverse of the rate can be expressed as lim n →∞ t n A ( t n ) = lim n →∞ ∑ n i =1 T i n = E [ T ] = 1 λ . P ( T > τ + t | T > τ ) = P ( T > τ + t and T > τ ) P ( T > τ ) = P ( T > τ + t ) P ( T > τ ) = e- λ ( τ + t ) e- λτ = e- λt As a direct application of the memoryless property, consider Z 1 to be the waiting...
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This note was uploaded on 03/27/2010 for the course CSCI 415 taught by Professor Saadmneimneh during the Spring '08 term at CUNY Hunter.

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Note10 - Computer Networks Modeling arrivals and service with Poisson Saad Mneimneh Computer Science Hunter College of CUNY New York Now remember

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