1. the Celebrated (and Demonized!) Poisson Proccess

2. 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)

3. 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

4. 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

5. 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 …

6. Poisson Distribution

7. 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)

8. 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)

9. 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)

10. 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)

11. 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

12. 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)

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

14. 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?

15. 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?

16. 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
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?

17. 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)

18. 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?

19. 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