Network Control Modelling and Analysis circuit-switched networks - - PowerPoint PPT Presentation

1/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Network Control Modelling and Analysis

Bruno Tuffin

INRIA Rennes - Bretagne Atlantique

PEV: Performance EValuation M2RI - Networks and Systems Track Rennes

2/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Outline

1

Introduction: goals and methods

2

Control in circuit-switched networks

3

Control in datagram networks

4

Control in virtual circuit switched networks

5

Conclusions

3/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Few references

1

• J. Walrand and P. Varaiya. High-Performance Communication
• Networks. Morgan Kaufmann Publishers, 2nd Edition, 2000.

2

• S. Keshav. An Engineering Approach to Computer Networking,

Addison-Wesley, 1997.

3

Ferguson P., Huston G. Quality of Service: Delivering QoS on the Internet and in Corporate Networks. John Wiley & Sons, Inc., 1998.

4

• F. Kelly. Models for a self-managed Internet. Philosophical

Transactions of the Royal Society A358, 2335-2348, 2000.

5

• L. Mamatas, T. Harks and V. Tsaoussidis. Approaches to

Congestion Control in Packet Networks. Journal of Internet Engineering, Vol.1, No.1, 2007.

6

Network Traffic Modelling and Control http://www.opalsoft.net/qos/ (very practical view).

4/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Introduction

Goal: describe concepts and techniques (from the modelling point of view) to control

Circuit-switched networks datagram (or packet-switched) networks (Internet) virtual circuit-switched networks (ATM).

A proper control mechanism allows to carry more traffic with the same quality of service (QoS). Also related to capacity planning. Different ways to operate:

admission control routing flow and congestion control resource allocation pricing (cf course on game theory and pricing).

Non-exhaustive view! Just few illustrations.

5/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Control methods

Different control methods at different time scales Admission control (for circuit or virtual circuit switched networks) determine which connection request can be accepted, depending on the network status. Can be delayed, or totally rejected. Routing decides the path from source to destination. Multicast or unicast can be envisaged. It can be dynamic or static. Congestion control (for datagram or virtual circuit switched networks), accelerates or slows down transmission according to congestion signals (ex: TCP). Can be seen as traffic shaping. Resource allocation (for virtual circuit switched networks) consists in controlling bandwidth and buffer allocated to a virtual circuit. This can also ba applied statically or dynamically. Pricing...

6/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

QoS

There is a wide range of QoS levels, from very small loss probability and delay, to Best Effort, with, in between, guaranteed bounded loss or delay. Example: in optical fibers: transmission error probability 10−10, and delay of 1 ms acceptable. Best effort: we do our best, without any guarantee, it depends on available resources. In between, you may ask for some loss probability and/or average or max delay requirements. Defining the level of desired QoS depends on the application (Video, Audio, email, file transfer have very different requirements), and how stringent the user is (wireless users less stringent than fixed telephony...) Service differentiation can be applied.

7/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Blocking in circuit-switched networks

Ex: telephone networks. QoS is the blocking probability (all lines busy). Designer problem: select the number of lines and switches, topology... If the network has K switches, L links, link i having Ci circuits. R routes, given by matrix A with Ar,i = 1 if route r goes through link i, 0 otherwise (fixed routing). Calls along route r: Poisson process with rate λr, and exponentially distributed with rate µr. State: n = (n1, . . . , nR) with ni number of active calls

• n route r.

State space X = {n| r

r=1 nrAr,i ≤ Ci

∀i}.

8/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Blocking and Loss models (2)

Product-form solution (to be verified from the balance equations) ∀n ∈ X: π(n) = 1 G

R

• r=1
• λr

µr

nr nr! with G normalizing constant, G =

n∈X

R

r=1

• λr

µr

nr

nr!

. From PASTA property, Br =

n:nr=Cr π(n).

Closed-form solution, but simulation needed for very large systems (complexity issue)!

For generalities on loss networks: K.W. Ross, Multiservice Loss Networks for Broadband Telecommunications Networks, Springer-Verlag, 1995. Online at

http://cis.poly.edu/~ross/LossNetworks/LossNetworks.htm.

For an example of protocol design and performance analysis:

• P. Dietrich and R.R. Rao. Request Resubmission in a

Blocking, Circuit-Switched, Interconnection Network. IEEE Transactions on Computers, Vol. 45, No. 11, 1996.

9/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Blocking and handoff in wireless (2G) networks

See G. Haring, R. Marie, R. Puigjaner, K.S. Trivedi. Loss Formulas and their Application to Optimization for Cellular Networks. IEEE Transactions on Vehicular Technology, 50(3):664-673, 2001. Exercice:

Consider a cell in a wireless (2G) cellular network. New calls arrive according to a Poisson process with rate λ1 and “handoff” calls, i.e., those coming from a change of cell but were already initiated, according to a Poisson process with rate λ2. Each call (new or handoff) ends with rate µ1, but each call can also leave the cell (due to mobility) , with rate µ2. The number of channels is limited to n. When a handoff call arrives and a channel is free, the call is accepted,

• therwise, it is lost. When a new call arrives, it is accepted only if at

least g + 1 cahcannels are available, otherwise it is blocked. This allows a better QoS, with priority to handoff calls. The transition diagram is given as follows, where state k means k

• ccupied channels.

n−g n k 1

10/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Handoff exercice: questions

1

What kind of process do we have?

2

Write the transition rates on the transition diagram.

3

Let λ = λ1 + λ2, µ = µ1 + µ2, A = ρ = λ/µ and A1 = λ2/µ. show that steady-state probabilities are πk = π0 Ak k! if k ≤ n − g, πk = π0 An−g k! Ak−(n−g)

1

if n − g + 1 ≤ k ≤ n, with π0 = 1

k=0 Ak k! +

k=n−g An−g k!

Ak−(n−g)

1

.

4

Using PASTA property, prove that the loss probability of a handoff call is Pd(n, g) =

An−g n!

Ag

1

k=0 Ak k! +

k=n−g An−g k!

Ak−(n−g)

1

.

5

From the same property, determine the blocking probability of a new call, i.e., that it is not accepted by the system. It is Pb(n, g) =

k=n−g An−g k!

Ak−(n−g)

1

k=0 Ak k! +

k=n−g An−g k!

Ak−(n−g)

1

.

6

From this, we can find the value of g that maximizes a global cost function.

11/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Routing in circuit-switched networks

In our previous model with K switches, L links and R routes, routes were fixed. Now routing can be selected to minimize costs or maximize revenue. Static routing:

There may be several route r from point A to B. Revenue W =

r wrλr(1 − Br), with wr revenue for

route r. Goal: to partition call rates λAB among routes r from A to B to maximize revenue. Optimization under contraints.

Adaptive routing:

Looks for an available route. Makes use of all available resource, but increases

• verhead.

12/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Routing in circuit-switched networks (2)

Dynamic Alternative Routing:

Takes congestion into account. Used in UK in the trunk network. Lessens the impact of forecasting errors, make better use of spare capacity and respond robustly to failures and overloads Fills an alternative route randomly chosen between each node pair, before a blocking occurs on that alternative

• route. In response to the blocking, a new alternative

route is randomly chosen (for future use) from the whole set. Decentralized and requires only local information to make routing decisions.

See also: Kelly, F.P. Network routing. Phil. Trans. Roy.

• Soc. Ser. A337 (1991), 343-367.

13/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Multiple Access in datagram networks

Decentralized view: game theory is the basic tool! We may also think that users do not play with

• parameters. Then how should they be fixed?

Typical applications:

CSMA (Wifi), ALOHA (medium access control)

• protocols. Sensing of channel is free, and sending after
• while. Collisions may happen if sending at the same
• time. How to decide when to send exactly?

Example: A.B. MacKenzie and S.B. Wicker. Stability of Multipacket Slotted Aloha with Selfish Users and Perfect Information. IEEE Infocom, 2003. Understand protocols and build more robust ones.

Channel given by a MultiPatcket Reception (MPR) matrix R =       ρ10 ρ11 . . . . . . ρ20 ρ21 ρ22 . . . . . . . . . . . . . . . . . . . . . . . . ρn0 ρn1 . . . ρnn . . . . . . . . . . . . . . . . . . . . .      

14/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

ALOHA model (2)

with ρnk probability that k packets successfullly received in a slot when n packets are transmitted. Probability of success then: k/n. Expected number of successes when n transmissions rn =

k=0 kρnk.

Stability condition: arrival rate less that limn→∞ rn. Probability of a particular user’s transmission:

k=0 ρnkk/n = rn/n.

A user has just one packet to send and can choose to transmit or to wait. Probability of n arrivals λn, expectation

k=0 kλk.

Value for success: 1. Cost per transmission c (0 if not transmitted). Ex: payoff for immediate success: 1 − c. Discount rate δ ∈ [0, 1), so expected payoff

t=t0 δt−t0ut when

entering at t0 and where ut immediate payoff at t. Strategy: transmission probabilities σ(t) ∀t. Theorem: there exists al least one equilibrium stategy σ.

15/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Routing in datagram Networks

In a network of M/M/1 queues, average delay faced by a packet along a path: D = 1

γ

• i

λi µi−λi

λi arrival rate at node i µi service rate at node i γ total rate arriving into the network.

Static routing for two queues in parallel: choose the probability p of sending to queue 1.

Minimize D = 1

λ

• λp

µ1−λp + λ(1−p) µ2−λ(1−p)

• .

Optimal p∗ can easily be found.

Static routing, general case:

λi = γi +

j λjpji with γi arrival rate from outside.

D in terms of the λis, themselves in terms of the pji. Optimization to be realized.

16/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Dynamic Routing in datagram networks

Do not select fixed probabilities to route packets, but devise a dynamic algorithm depending on actual backlogs. For queues in parallel: given state n = (ni)i for queues lengths, optimal p(n)? Still beyond the reach of current approaches. Approximations (heuristics).

Ex: Bellman-Ford Algorithm. Estimates the average delay on each link dij between i and j. Let xi be the minimum delay between i and some fixed fixed destination xi = min

j (dij + xj).

Fixed-point equation x = F(x), solved by recursion xn+1 = F(xn). Once x known, fastest path indentified by sending to the node with minimum value.

17/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Flow control in datagram networks

Open loop:

each source describes its traffic parameters and resources (buffern bandwidth...) are reserved accordingly. During transmission, the source can shape its traffic. Network overload avoided.

Closed loop:

the source dynamically adapts its flow. changes in the network provided by explicit or implicit

• the source measures itself) feedback schemes.

18/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Kelly et al’s work: very popular in recent years

Decentralized efficient control of networks, and Internet Modelling. Elastic traffics where the rates are proportional to the willingness to pay of each user.

F.P. Kelly. Mathematical modelling of the Internet. In Proceedings of the Fourth International Congress on Industrial and Applied Mathematics, 2000. F.P. Kelly. Charging and rate control for elastic traffic. European transactions on Telecommunications, 8:33–37, 1997. F.P. Kelly, A.K. Mauloo, and D.K.H. Tan. Rate control in communication networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society, 49:237–252, 1998. S.H. Low and D.E. Lapsley. Optimization Flow Control, I: Basic Algorithm and Convergence. IEEE/ACM Transactions on Networking, 7(6), 1999. S.H. Low, F. Paganini, and J.C. Doyle. Internet Congestion Control. IEEE Control Systems Magazine, 2002.

19/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Basic model

Consider a set of J resources with capacity Cj for resource j. Again, route r is a non-empty subset of J and R set of possible routes. Let Ajr = 1 if j ∈ r and 0 otherwise and define A as A = (Ajr). Ur(xr) utility function of user r when the flow rate is xr. Ur is assumed to be an increasing, strictly concave and continuously differentiable function. U = (Ur(·), r ∈ R) and C = (Cj, j ∈ J). Centralized goal: (GLOBAL PROBLEM) maximize

• r∈R

Ur(xr) subject to Ax ≤ C and x ≥ 0.

20/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Dealing with user’s point of view

From the user point of view, maximize over ωr ≥ 0 (USER PROBLEM) Ur ωr λr

• − ωr,

the flow rate is xr = ωr/λr ωr is the amount that user r is willing to pay per unit time λr is the charge per unit flow and unit time for user r.

Assume that the network knows ω = (ωr, r ∈ R) and attempts to maximize (NETWORK PROBLEM)

• r∈R

ωr log xr subject to Ax ≤ C and x ≥ 0.

Very convenient: allows to compute optimal flow rates very easily.

21/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Solution

There always exist vectors λ, ω and x satisfying ωr = xr/λr ∀r ∈ R such that

ωr maximizes USER PROBLEM x maximizes NETWORK PROBLEM and then x is the unique solution maximizing GLOBAL PROBLEM.

Vector of rates x per unit charge is propotionally fair: if x ≥ 0 and Ax ≤ C, and for any other feasible vector x∗, the aggregate proportional change is zero or negative

• r∈R

ωr x∗

r − xr

xr ≤ 0.

22/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Decentralized view

Maximization of NETWORK PROBLEM centralized, indesirable. To proceed in a decentralized way:

Consider the system of differential equations d dt xr(t) = κr  ωr(t) − xr(t)

• j∈r

µj(t)   where µj(t) = pj

• s:j∈s xs(t)
• shadow price per unit

flow through j and pj(t) derivative of the rate at which cost in incurred at resource j when load is y. Motivation:

resource j generates a continuous stream of feedback signal at rate ypj(y) when total flow y; resource j sends a proportion xr/y of these feedback signals to a user r (with a flow of rate xr) user r views each feedback signal as a congestion indication requiring some reduction of flow xr.

The system has a unique value x such that xr = ωr/

j∈r µj arbitrarily closely approximates the

• ptimization of NETWORK PROBLEM.

23/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Kelly control and TCP

In TCP, congestion indication is from dropped or marked packets. Though, shown that TCP can be modeled by the system of differential equations d dt xr(t) = mr T 2

r

mr T 2

r

+ xr(t)2 2mr

j∈r

µj(t) and acting as if the utility function of user r is √ 2mr Tr arctan xrTr √ 2mr

• Tr is the round trip time for the connection of user r

mr is a parameter which would inter alia multiply by m the rate of additive increase and make 1 − 1/2m the multiplicative decrease factor in Jacobson’s TCP algorithm. Stable point with pr =

j∈r pj, xr = mr

Tr 2(1 − pr) pr 1/2 .

24/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Low et al’s work

Main difference: users decide their rates and pay for it whereas in Kelly et al.’s work, users decide their payment and receive what the network allocates. Set L of unidirectional links of capacities cl, l ∈ L. Set S of sources characterized by utility fuction Us(xx) concave, increasing in its transmission rate xs Goal: maximize

s∈S Us(xs) over xs subject to

capacity constraints. Decomposed in:

1

Each link receives the rates xs(t) if s’s route is through link l.

2

Each link l calculates its price pl(t + 1) for a unit of bandwidth (in order to optimiwe the benefits) using the gradient projection algorithm pl(t + 1) = [pl(t) + γ(xl(t) − cl)]+ where γ is a stepsize.

3

Each link communicates pl(t + 1) to each source whose route is through link l.

25/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Low et al.’s work (2)

Then the algorithm for each source is:

1

Each source is fed back the price ps =

L(s) pl where

L(s) is the set of links that s uses.

2

The source chooses then its transmission rate xs (in an interval (ms, Ms]) which maximizes its benefit Us(xs) − ps(t)xs.

3

These rates xs(t + 1) are send to the links which calculate again new prices and so on.

26/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Modelling TCP by AIMD processes

TCP Modelling very important issue, especially at the beginning of the decade. Basic version of TCP: increases linearly its transmission rate up to congestion notification. Multiplicative decrease then. Many models exist. First example, based on

• E. Altman, D. Barman, B. Tuffin and M. Vojnovic.

Parallel TCP Sockets: Simple Model, Throughput and

• Validation. In IEEE INFOCOM, Barcelona, March 2006.
• E. Altman, E. El-Azouzi, D. Ros and B. Tuffin. Loss

Policies for Competing TCP/IP Connections. Computer Networks, Vol.50, Num.11, pages 1799-1815, 2006.

Additive Increase Multiplicative Decrease (AIMD) model. N TCP sessions competing for bandwidth. Capacity C of the bottleneck router (service rate; no buffer).

27/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

TCP and AIMD processes

ηi (β(i)) additive increase rate (multiplicative decrease) for session i

values depend on the RTT ηi = 1/RTT 2

i (window size increases by one each RTT;

throughput given by the window size/RTT, leading to the result).

Tn n-th congestion epoch; τn+1 = Tn+1 − Tn Y (i)

n

throughput of session i before the n-th congestion epoch a(i)

n = 1 if session i experiences a loss, 0 otherwise

Let γ(i)

n

= (1 − a(i)

n ) + β(i)a(i) n , multiplicative value

effectively applied. Then Y (i)

n+1 = γ(i) n Y (i) n

+ τn+1ηi. A loss is experienced as soon as N

i=1 γ(i) n Y (i) n

+ τn+1 N

i=1 ηi = C (No buffering).

p(i) = E[a(i)

n ]; i p(i) ≥ 1.

28/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Insnsitivity result for ONE loss in the symetric case with β = 1/2

If Fixed propability loss, p(i) = 1/N, the equation can be solved and ¯ Y (1) =

• 1 −

1 1+N(1+β)/(1−β)

• C.

In the more natural case of proportional loss, p(i)

n

=

Y (i)

n

j=1 Y (j) n

= Y (i)

n

C , ¯

Y (1) =

• 1 −

1 1+N(1+β)/(1−β)

• C.

For Largest Throughput Loss (LTL), ¯ Y (1) =

• 1 −

1 1+N(1+β)/(1−β)

• C;

In general, ¯ Y (1) =

• 1 −

1 1+N(1+β)/(1−β)

• C.

On the other hand, the variance of flow is not the same (throughput variability). If N = 2,

Fixed: 5C 2/24, Proportional: 679C 2/3396, LTL: 4C 2/21.

29/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Numerical comparison for the fairness ¯ Y1/ ¯ Y2

5 10 15 20 25 30 35 1 2 3 4 5 6 7 Ratio of throughputs Ratio of RTTs y = x2 (3x2+5)/(5x2+3) 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1 2 3 4 5 6 7 Ratio of throughputs Ratio of RTTs

The asymptotic value 5/4 for LTL can be proved. Example: when the ratio of RTT is 3, the ratio of throughputs is 1.21 for LTL, 2.75 for the proportional strategy and 6 for the constant strategy.

30/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

TCP throughput

Jitendra Padhye, Victor Firoiu, Don Towsley, Jim

• Kurose. Modeling TCP Throughput: A Simple Model

and its Empirical Validation. In Proceedings of SIGCOMM’98, 1998. Analytic characterization of the steady state throughput as a function of loss rate and round trip time for a bulk transfer TCP flow. ”triple-duplicate” (TD) acknowledgments, i.e., four ACKs with the same sequence number. Focus on the congestion avoidance behavior of TCP and its impact on throughput. A round starts with the back-to-back transmission of W packets and ends with the reception of the first ACK for

• ne of these W packets.

31/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Assumptions and notations

Assume that

each user has an unlimited amount of data to send. time spent in slow start negligible compared to connection length. Time needed to send all the packets in a window is smaller than the round trip time. Losses are independent between rounds. Packet losses correlated among back-to-back transmissions within a round. If a packet lost, so are all remaining packets until the end of that round.

TD period: period between 2 TD losses. For the i-th TD period, TH = E[Y ] E[A] , with

Yi number of packets sent within i-th period Ai duration of i-th period Wi window size at the end of i-th period. αi the first packet lost in TDP i Xi round where this loss occurs p loss probability of a packet b number of packets acknowledged by a received ACK (i.e., delayed ACK, typically 2).

32/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Analysis

I P[α = k] = (1 − p)k−1p, E[α] = 1/p. Then E[Y ] = E[α] + E[W ] − 1 = 1−p

p

+ E[W ] Rij RTT of i-th round of TDP i, then Ai = Xi+1

j=1 Rij.

Since Rij independent of congestion window, that is of j E[A] = (E[X] + 1)E[R] = (E[X] + 1)RTT. Wi = Wi−1

2

+ Xi

b , leading to E[W ] = 2E[X]/b.

If δi number of packets sent in the last round, Yi packets transmitted in TDPi Yi = Xi/b−1

k=0

• Wi−1

2

+ k

• b + δi.

Yi = XiWi−1

2

+ Xi

2

• Xi

b − 1

• + δi =

Xi 2

• Wi−1

2

+ Wi − 1

• + δi.

1−p p

+ E[W ] = E[X]

2

E[W ]

2

+ E[W ] − 1

• + E[δ].

If δi uniformly ditributed between 1 and Wi, E[δ] = E[W ]/2.

33/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Result

E[W ] = 2+b

3b +

• 8(1−p)

3bp

+ 2+b

3b

2, Leading to E[X] = bE[W ]/2 = 2+b

6

+

• 2b(1−p)

3p

+ 2+b

6

2 And E[A] = RTT

• 2+b

6

+

• 2b(1−p)

3p

+ 2+b

6

2 + 1

• .

From TH =

1−p p +E[W ]

E[A] , we get the relation... and eventually TH = 1 RTT

• 3

2bp

34/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Control in virtual circuit switched networks

Specific example: ATM networks. Contract in which nt=etwork guarantess specific QoS

• bounds. Requires measurements.

Deterministic procedures: leaky bucket (worst case analysis)

• r statistical ones.

35/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Leaky bucket : admission control at entrance

Consider a system with arrivals following a Poisson process with rate λ. Token mechanism to limit peak rate and throughput. To enter the system, a packet has to obtain a token. If

• ne available, fine, otherwise, lost.

Tokens stored in a queue with limited capacity B (say here B = 3 for exercice below). Every T time units, a token is generated. If the token queue is full, it is destroyed, otherwise, stored. That way:

max throughput: the one of the generation of tokens Peak rate: no more than B packets pass between two generation of tokens.

36/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Leaky bucket : Exercice

Let αn probability that n packets arrive in a slot of length T, αn = (λT)n

n! e−λT, and βn = ∞ k=n αk.

Let tn instants when new tokens are generated, tn = nT. A slot: interval [tn, tn+1). Xn number of tokens in the token queue at tn (between 1 and B).

1

Show that (Xn)n≥0 is a Markov chain, irreducible, aperiodic with steady-state regime.

2

Compute steady-state probabilities (πk)k.

3

Can probabilities πk be considered as proportions of time tfor which the token queue contains k tokens in steady-state?

4

What is the mean number of destroyed tokens per unit

• f time?

5

Deduce the entrance rate λe of packets and probability Pr of rejecting a packet.

37/37 Game Theory Bruno Tuffin Introduction: goals and methods Control in circuit-switched networks Control in datagram networks Control in virtual circuit switched networks Conclusions

Conclusions

Control can be performed in many different ways (admission, congestion...). Very active issue. We could not deal with all aspects of control, just wanted to give you some hints. Modelling and performance evaluation valuable tools for protocol improvements... Then, can be verified by simulation tools such as NS2 (or NS3). If you look at high-level journals or conferences, there almost always is at least one modelling and performance evaluation section.