the Celebrated (and Demonized!) Poisson Proccess

Arrivals of Customers/Packets: How to Model? Iceland Volcano: Why could you not talk to airline cust. service? New Years Eve: Why can I not call my relatives? Problem 1: Call Center Dimensioning Customers call randomly Assume (for now!) duration of each call is fixed N workers : if all busy, call is dropped Question: What should N be to ensure at most 5% of calls are dropped? Case 1: calls arrive regularly (one every X min) Case 2: calls arrive in bursts (many together, then silence)

Arrivals of Customers/Packets: How to Model? Problem 2: Internet Router Buffer Sizing Packets arrive at a core router Need to be buffered before forwarded further Question: How large should the buffer be (to ensure few drops)? buffers

Lesson: Arrival Models => System Design Need to know/model the (random) arrival of “work” => to optimize the system! Calls at a call center => to pick the number of employees Calls to a base station (inside a cell) => to allocate frequencies Packets at a router => to choose the right buffer size (large) jobs at a cluster/supercomputer => to choose the number of CPUs What might we need to know? Average amount of work per min/hour/day Probability of 3, 4, 5 customers arriving within T min Probability that > N customers arrive within T min

Poisson Distribution Rate of events: λ average number of events in an interval Probability of n events in an interval Examples well approximated by Poisson distribution The number of deaths per year in a given age group. The number of phone calls arriving at a call centre per minute. The number of new sessions arriving at a web server per hour The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry …

Poisson Distribution

Poisson Process (Definition 1) time Definition 1: A counting process viewpoint Property 1 (“independent increments”): # of arrivals in non-overlapping intervals (e.g. N(T1) and N(T2)) is independent Property 2 (“stationary increments”): # of arrivals in [t1,t2] only depends on (t2-t1) Property 3: # of arrivals N(t) in interval t is Poisson (λt)

Poisson Process: 2 More Definitions T: exponential Definition 2: a Renewal Process viewpoint Inter-arrivals times are independent Time T between arrivals (“renewals”) is exponential(λ) dt Definition 3: “aggregate of many rare events” Prob{1 event in dt} = λdt + o(dt) (independently of past events) Prob{> 1 events in dt} = o(dt) (negligible as dt -> 0)

All Definitions are Equivalent!! We can go (prove) from any definition to any other Definition 1 (Poisson) => Definition 2 (Exponential) Prob{T > t) = Prob{0 events in t} => T is exponential Definition 1 (Poisson) => Definition 3 (rare events)

Poisson as a Binomial Approximation P{arrival} = λ•δt Number of arrivals N(t) in t  Binomial(n, p) n = t/ δt p = λ•δt + o(δt) If δt  0, such that np = λt: Then Binomial (n,p)  Poisson (λt)

Poisson Properties: Waiting Time to n-th Arrival Time to wait until the next arrival (T1) is exponential Time to wait until the n-th arrival (Sn=T1+T2+…+Tn)? Sum of n independent and identically distributed (IID) exponential random variables Gamma Distribution How to get this? Proof 1: Moment Generating Function Proof 2: (CDF) Fs(t) = Prob{Sn ≤ t} = P{N(t) ≥ n} (Ross, Ch.2) Proof 3: P{t < Sn < t+dt} = P{n-1 events in t,1 event in (t,t+dt)} (Ross) T1 T2 t Sn

Poisson Properties: 1 arrival in a window T 1 arrivals in T S1 t T We are told that 1 arrival has occurred in the interval T Question: When did it happen exactly? NOTE: this is a conditional probability Answer: Arrival is uniformly distributed: any instant in the interval is equally probable P{S1 ≤ s} = s/T (0 ≤ s ≤ T)

Poisson Properties: N arrivals in a window T t n arrivals in T N arrivals in T  each arrival is uniformly distributed

An Example: Energy-Efficiency Sensed data: Poisson (λ) t T 2T sleep wakeup wakeup Receives event readings with rate λ  must sent to a base station To save battery power: (a) wireless card in sleep mode, (b) queue events during sleep mode, (c) wake up every T minutes and transmit all queued events QoS: When an event is queued for  cost of queueing for t : c(t) = ct Q: What is the total cost incurred each period T? A: 0.5 • c •λT2 Q: Assume battery consumption is a(T) = a/T. What is the optimal T?

Poisson Thinning/Sampling Assume Poisson arrivals with rate λ A 2nd random process is created as follows: We accept each arrival with probability p < 1 (or reject with 1-p) X accept with prob p T Question: what is the expected number of arrivals within T? Answer: p•λT Question: what is the second process? Answer: Poisson with rate pλ Proof?

Poisson Thinning Examples Load Balancer: Assign job i with probability pj to CPU j Q: Input process to CPU j? A: Poisson with rate pj ·λ scheduler www.movie-clips.com: 1 slow and 1 fast server| slow fast Poisson rate λ job size < S e.g. short clips CPU1: p1 CPU2: p2 CPU3: p3 CPU4: p4 load balancer New jobs: Poisson rate λ job size ≥ S e.g. long movies Job size x is random ~ CDF is F(x) Q: Is the input process to the slow server Poisson?

Poisson Process Merging Ethernet packets from each Base Station are Poisson Poisson λ1 Poisson λ2 Poisson λ3 Question: what is the arrivals process of ALL packets at the input of the Ethernet switch? Answer: Poisson with rate λ1 +λ2 + λ3 rate λ1 + rate λ2 Poisson(λ1+λ2)

Compound Poisson Process Map of open WiFi access points (AP) Definition: A stochastic process [X(t), t ≥ 0] is a compound Poisson process if: [N(t), t ≥ 0] is a Poisson process [Yi, i ≥ 1] is a family of IID random variables, independent of N(t) Results 1) E[X(t)] = λt•E[Y1] 2) Var(X(t)) = λt•E[Y12] User uploads (a large no. of) pictures on DropBox using WiFi only User walks around  randomly encounters APs as a Poisson process with rate λ Bytes uploaded during each WiFi is random: Yi Depends on speed, congestion, distance Q: How long until all pictures uploaded?

Poisson Process: Why We Like It Memory-less property: simplifies models No need to know/keep track of the past to predict future Stationary behavior is sufficient! Good approximation for aggregate “traffic” of many and independent sources Palm-Khintchine Theorem Why we don’t like it: Not always true Many workloads have “heavy-tailed” properties  memory