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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 interarrival 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 interarrival 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.
 Spring '08
 SaadMneimneh
 Computer Networks

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