Lecture 7 – November 11, 2020 Wi Wireless Access Graduate course in Communications Engineering University of Rome La Sapienza Rome, Italy 2020-2021
Poisson processes
Poisson random process • Reference tool in modelling the distribution of discrete events in time • Can be generalized to the birth and death process • Examples: – Arrival of messages (packets) at a digital communication multiplexer – Arrival of photons in a light beam at an optical receiver (optical communications)
Poisson random process • Definition of Random points in time : points in time at which an event occurs • Let the time of the k -th arrival be denoted by t k where t k ≥ t j for k > j • Define a continuous-time random process N(t) equal to the number of arrivals since starting time t 0 up to current time t . • We call N(t) a counting process
Poisson random process • N(t) has the following properties: – it only assumes non-negative values; – it has initial condition N(t 0 ) = 0; – it increases by 1 at each random point in time. b)Arrivals and departures a) Only arrivals
Poisson random process b)Arrivals and departures • Typical example: a queueing system • N(t) is the difference between number of arrivals and number of departures accumulated over time • The model is adequate for example for a buffer in a packet- switching network
Poisson random process • In most cases, the count N(t) is all we need to predict the system evolution after t • When this is true, N(t) is the state of the system at time t : we say that the system is in state j if N(t)=j • The system can change state at any time (differently from what we saw for Markov chains) • Note that a sample N(t i ) of the counting process at time t i is a discrete-valued random variable • We define the probability of being in state j as: ( ) = Pr N t = t i ( ) = j ⎡ ⎤ ∀ t i ∈ R q j t ⎣ ⎦
Birth and death process • We need to model the evolution of the system from one state to the other • We cannot use the probability of transition from one state to the other at a given time t , because it will be almost always equal to 0 • We can define, however, the rate R of transitions between two states • If R is a constant, in a time δ t one has R δ t transitions • For δ t 0 , we can neglect the possibility of two transitions in δ t , i.e.: – R δ t is the probability of one transition in δ t, – (1 - R δ t) is the probability of no transition in δ t.
Birth and death process • We can thus define the following state transition diagram for a birth and death process: where: ( ) , • λ i t i = 0,1, are the transition rates from state i to state i+1 ( ) , • µ j t j = 1, 2, are the transition rates from state j to state j-1
Birth and death process • The evolution of the process is then governed by the following set of equations: Birth from j-1 Birth to j+1 ( ) ⎧ ( ) q j t dq j t ( ) q j − 1 t ( ) + µ j + 1 t ( ) q j + 1 t ( ) − λ j t ( ) + µ j t ( ) ( ) , = λ j − 1 t j ≥ 0 ⎪ ⎨ dt ⎪ ( ) = 0 Death from j+1 Death to j-1 q − 1 t ⎩ with the conditions: ( ) = 1, ⎧ ⎪ q 0 t 0 ⎨ ( ) = 0 ∀ j > 0 ⎪ q j t 0 ⎩
Pure birth process • The special case of a birth and death process with: ( ) = λ , λ i t i = 0,1, ⎧ ⎪ ⎨ ( ) = 0, µ j t j = 1, ⎪ ⎩ is known as pure birth process or Poisson process with constant rate λ • For this process one has the following diagram: • and the following set of equations: ( ) dq j t ( ) = 1 ( ) − λ q j t ( ) , with = λ q j − 1 t j ≥ 0 q 0 0 dt
Pure birth process • The first order differential equations can be solved easily • If we define the Laplace transform of the state probability: ∞ ( ) = ( ) ∫ e − st dt Q j s q j t 0 and take the Laplace transform of both sides we get: ( ) − q j 0 ( ) + λ Q j s ( ) = λ Q j − 1 s ( ) sQ j s • Taking into account the initial condition in t 0 =0 one has: λ 1 ( ) = ( ) = ( ) , and j > 0 Q 0 s Q j s s + λ Q j − 1 s s + λ • Which can be solved by iteration leading to: λ j ( ) = Q j s ( ) j + 1 s + λ
Pure birth process • And finally, by taking the inverse Laplace transform, one has, for t ≥ 0 and j ≥ 0 : Poisson distribution ( ) ⎦ = λ t j ( ) = Pr N t ( ) = j j ! e − λ t ⎡ ⎤ q j t ⎣ • Thus a pure birth process is actually a Poisson process with constant rate (but the result can be generalized for a variable rate as well)
Poisson distribution properties • The Poisson distribution with parameter a ( ) k ( ) = a k ! e − a p N k has the following properties: 1. A random variable N characterized by a p.d.f. equal to a Poisson distribution with parameter a has the following expectations: [ ] = a Mean E N 2 = a [ ] ⎡ ⎤ ⎦ − E N Variance E N 2 ⎣ 2. The moment generating function for the random variable N takes the form: ( ) ( ) = a e s − 1 log e Φ N s
Poisson process • Going back to the Poisson process, it can be proved that, for a general initial condition N(t 0 )=k , one has: ( ) ( ) j − k ( ) = λ t − t 0 ( ) , for j ≥ k , t ≥ t 0 − λ t − t 0 q j t e ( ) ! j − k – which is a Poisson distribution with parameter λ (t-t 0 ) , that is the expected number of arrivals from instant t 0 . • (j-k) is the count since the starting time t 0 • IMPORTANT: the number of arrivals starting from t=t 0 has a distribution that does not depend in any way on what happened before t 0 • As a consequence, the number of arrivals in the interval (t 0 ,t) is statistically independent of the number of arrivals in any other non-overlapping interval of time
Pure death process • The special case of a birth and death process with: ( ) = 0 ⎧ λ i t ⎪ ⎨ ( ) = j µ µ j t ⎪ ⎩ is known as a pure death process , and is characterized by the following transition diagram: • Under the assumption that the state of the system in t=0 is n , it can be proved that for this process one has the following state probabilities: (Binomial ⎛ ⎞ ( ) = e − µ t ( ) = n ( ) ( ) 1 − p t ( ) n − j ⎟ p j t with p t q j t distribution) ⎜ ⎝ ⎠ j
Poisson process with variable rate • Let us consider a Poisson process with time-varying rate, that is λ = λ (t) • The set of equations describing the system is: ( ) ⎧ dq j t ( ) q j t ( ) = λ t ( ) q j − 1 t ( ) , + λ t j ≥ 0 ⎪ ⎨ dt ⎪ ( ) = 0 ⎩ q − 1 t with: ( ) = 1 ⎧ ⎪ q 0 t 0 ⎨ ( ) = 0 j ≥ 1 ⎪ q j t 0 ⎩ assuming thus that the system is in state j=0 at t=t 0
Poisson process with variable rate • Since λ (t) depends on time t the Laplace transform is of no help • One defines then: t ∫ ( ) = Λ t λ τ ( ) d τ t 0 that is the total number of arrivals in the interval [t 0 , t] : Λ (t) λ λ (t) λ t
Poisson process with variable rate • The probability of n arrivals in the interval [t 0 , t] is then determined by a Poisson distribution with parameter λ (t) ( ) ( ) = Λ n t ( ) −Λ t q n t e n ! in perfect analogy with the constant rate case • In this case one has for the number N(t) of arrivals during the interval [t 0 , t] the following expectations: ( ) ( ) ⎦ = Λ t ⎡ ⎤ Mean E N t ⎣ ( ) σ N t = Λ t 2 Variance ( )
Birth and death process: the queueing problem • Birth and death processes are particularly suitable for modelling queueing systems • We will adopt the following convention/terminology: – A queue is a buffer or memory which stores messages/packets – Messages in the queue are cleared by one or more servers ( each server processes one packet at the time) – The queue can only keep a limited number of messages, referred to as the number of waiting positions – The state of the system (which tracks a counting process) is the sum of the packets waiting in the queue and of the packets currently being served – Messages arrive at the queue ( births ) at random times, and depart from the queue ( deaths ) due to the completion of service
The queueing problem Example: one server and no waiting positions • Constant arrival rate λ • Constant service (departure) rate μ • Since there is one server and no waiting positions, if a message arrives while the server is busy the message is lost -> only two states, 0 and 1 • It can be proved that the probability that the server is not busy is: µ ⎡ µ ⎤ ( ) = ( ) − ( ) t − µ + λ λ + µ + q 0 0 q 0 t ⎥ e ⎢ λ + µ ⎣ ⎦ • In such a system is usually more interesting to derive the state probabilities at steady state, that is out of the initial transient
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