Delay models in data networks

1. Shanghai Jiao Tong University Delay models in data networks

2. Data networks and Queueing R R R R R R R S

3. General Methodologies of Queueing Analysis • We are given: • Packet arrival behavior • Packet length distribution • Packet routing / handling policies • We want to deduce: • Packet delay • Queue length • Packet loss • Queueing theory can also be applied in other areas, such as in analyzing Circuit Switched Net.

4. In this chapter • Poisson process • The Little’s Theorem • M/M/x Queueing systems • Burke’s Theorem and Jackson’s Theorem • M/G/1 • Reservation systems and priority queue

5. Weiqiang Sun Shanghai Jiao Tong University Arrival model and the little’s theorem

6. The arrival process • The arrival process can normally be described • by the number of arrivals in a unit time • or can be described by inter-arrival time • Poisson process • the most commonly used arrival model in telecom network • Named after the French Mathematician Simeon-Denis Poisson (1781 – 1840)

7. Examples of Poisson process • The number of page request arriving at a web server (no attack, please) • The number of telephone calls arrives at an switch • The number of photons hitting a photon detector, when lit by a laser • The execution of trades on a stock exchange • …

8. Three ways to define a Poisson process (1) In an infinitesimal time interval dt, there may occur only one arrival, and this happens with probability λdt dt

9. Three ways to define a Poisson process (2)The number of arrivals N(t) in a finite interval of length t • Obeys Poisson distribution with parameter λt • The number of arrivals in non-overlapped intervals are independent T1 Poisson(λT1) T2 Poisson(λT2)

10. Three ways to define a Poisson process (3) The interval times are independent and obey exponential distribution with rate λ exp(λ) • Proof of 23

11. Properties of Poisson process • 1st of all: memory-less • The additional time to wait is independent on when it starts • P(X>40|X>30) = P(X=10) • Think about the coin-tossing process, though not Poisson, it is memory-less • X is the number of trials until the first “head” • Additional trials to get a “head” is independent of previous trials

12. Properties of Poisson process (cont.) • Merging property • Let A1, A2, …, Ak be independent Poisson process of rate λ1, λ2,…, λk, A= ∑Ai is also Poisson with rate λ= ∑ λi λ1 λ1 ∑ λi λ2 λ2 λ1+λ2 λk

13. Properties of Poisson process (cont.) • Selection property • Suppose a random selection is made from a Poisson process (λ), each arrival is selected with probability p, independent of the others, the resulting process is a Poisson process with rate pλ λ1 λ ∑ λi λ2 pλ λk • Splitting property • The above property also leads to random splitting property, why and how?

14. Properties of Poisson process (cont.) • PASTA: Poisson Arrival See Time Average • One of the central tools in queueing theory • An arrival customer always see the system in average state, in terms of number of customers in the system πi: the probability that an outside observer sees the system in state Si at a random instant πi*: the probability that an arriving customer sees the system in state Si just before arrival λ Si In general, πi ≠πi*. But for Poisson arrivals, they equal. To prove, show that:

15. Examples • Problem in Text, 3.6 • Problem in Text, 3.10(d)

16. Data networks and Queueing R R R R R R R S

17. Little’s Theorem • Named after John Little, an MIT Sloan prof. Little J. D. C. “A proof of the Queueing Formula L= λw,” Operation Research, 9, 383-387 (1961) λ A queueing system (N, T) • N= λT • λ: arrival rate of customers into the system • N: number of customers in the system • T: average amount of time a customer spends in the system

18. Some observations of Little’s Theorem • The result is very useful because of its generality • Nothing is assumed about the system • Can be applied to the whole system, or • Any part of the system • Treat system as a blackbox • The arrival process can be anything • Not necessarily Poisson process • But, it has to be stationanry • And it can naturally explain why • On a rainy day, traffic moves more slower and the streets are more crowded • A fast-food restaurant needs a smaller waiting room

19. A simple justification of Little’s Theorem • N(t) the number of customers in the system • N: average number of customers in the system, can be calculated by dividing the above shaded area by t • T: on average, each customer contributes T • the average number of arrivals during t is λt • Thus the area is λt×T, hence N = λT • N(t) N • 0 • t Graphical proof, see text 3.2.1

20. Application examples R 1. A transmission system R R R queue transmitter transmission line • 2. A complex system with multiple streams R λ3 R R λ2 λ1 λ1 λ2 λ3

21. Single server queues queue λ customers per second • M/M/1 • Poisson arrivals, exponential service times • M/G/1 • Poisson arrivals, general service times • M/D/1 • Poisson arrivals, deterministic service time (fixed) μcustomers served per second S

22. Discrete-time Markov chains • The memory-less property of both arrival process and service time • allow us to use the Markov chain theory to analyze M/M/1 queueing systems • Discrete-time Markov chains p0,1 p1,2 p2,3 pk-1,k pk,k+1 S0 S1 S2 Sk p1,0 p2,1 p3,2 pk,k-1 Pk+1,k pi,j : probability of state transition from i to j Si : states, i=0,1,…

23. M/M/1 systems and Markov chain • Define state k  k customers in the system • p(i, j): the probability that number of customers in the systems changes from i to j, within a very small time interval δ • It can be shown that the probability of more than one arrival / departure is o(δ) • Hence as δ0, we have: p(0, 0) = 1 - λδ p(j, j) = 1 – λδ – μδ, for j > 0 p(j-1, j) = λδ p(j+1, j) = μδ p(i, j) = 0, for |i-j|>1

24. Markov chain for M/M/1 systems • In equilibrium, the transition from state n to n+1 is the same as the transition in the reverse direction • λp(n) = μp(n+1) for all n • Local balance equations between two states (n, n+1) • p(n+1) = (λ/μ)p(n) =ρp(n), ρ=λ/μ p(n) =ρnp(0) • By axiom of probability: λδ λδ λδ λδ λδ 0 1 2 K μδ μδ μδ μδ μδ

25. Some results of M/M/1 • Average number of customers in the system: N • The average amount of time a customer spends in the system can be derived from the Little Theorem • The average amount of time a customer waiting in queue • Average number of customers in the queue

26. The example of circuit switching vs. packet switching Packet switching (multiplexing) Circuit switching (dedicated) T = ? • T = ? queue λ/M λ/M λ/M λ/M μ … … λ/M λ/M

27. m server systems: M/M/m • Departure rate is proportional to the number of servers in use μ queue λ μ …. m servers μ customer per second, per server μ λδ λδ λδ λδ λδ 0 1 2 m 2μδ 3μδ mμδ mμδ μδ m+1

28. M/M/m systems • Local balance equations • Solve for p(0) and p(n) using ∑p(n)=1

29. The Erlang C formula and other results • And the average number of customers in queue • The probability of being queued  the Erlang C formula • With the Little’s Theorem, average time in queue and in system • And of course, the average number of customers in system

30. M/M/m example • Text problem 3.7 M/M/2 systems with heterogeneous servers

31. M/M/∞ (M/M/inf) • Infinitive number of servers  Customers will no longer experience queueing delays 2μδ 3μδ mμδ (m+1)μδ μδ • Local balance equations λδ λδ λδ λδ λδ 0 1 2 m m+1

32. M/M/m/m • Same as M/M/m, but there is no queue • M/M/m no queue version • Customers who arrive finding all server busy will leave (they are blocked) • Blocking probability • The probability that a customer will come and find all server in service

33. M/M/m/m systems • Up to m customers in the system 2μδ 3μδ mμδ μδ • Local balance equations λδ λδ λδ λδ 0 1 2 m • The probability that a customer finds the system busy, the Erlang B Formula

34. The Erlang B formula • Define • The systems load in Erlang • Formula sensitive to the ratio of λ and μ • Can be used to dimensioning network capacity • Given tolerable PBand the load, find the number of server needed