introduction to network performance and queuing theory

Introduction to Markov Chains, Queuing Theory,

and Network Performance

Marceau Coupechoux

Telecom ParisTech, departement Informatique et Reseaux

[email protected]

IT.2403 Modelisation et Performance des Reseaux

February 8, 2011

Outlines

1. Network Performance

2. Mathematical Background

3. Markov Chains

4. Formalism of Queuing Theory

5. Simple Queues

6. Erlang B and C laws

7. Networks of Queues

51

Formalism of Queuing Theory

1. Arrival process and service time

2. Buffer mechanism

3. Kendall notation

4. Multi-class queues

5. Networks of queues

6. Performance parameters

7. Stability

8. Little law

9. Distributions at arrival and departure instants

52


Simple Queue

customers

Arrival process

Buffer

ServerDeparture of

Arrivals

• How do customers arrive ?

• Example: the inter-arrival times are constant

• Simple case: the inter-arrival times between customers are iid random variables

• Poisson arrivals: inter-arrival times are exponentially distributed

53


Simple Queue

customers

Arrival process

Buffer

ServerDeparture of

Service time

• How are customers served ?

• Are there more than one server ?

• How much time will the service take (service time distribution) ?

• Simple case: service times are iid random variables

• Exponential service: service time is exponentially distributed

54


Simple Queue

customers

Arrival process

Buffer

ServerDeparture of

Buffer mechanism

• How customers are stacked in the buffer and how customers are taken out of the bufferby the server

• FIFO: first in first out = FCFS : first come first served

• LIFO: last in first out

• RANDOM

• Round Robin: cyclic mechanism

• etc

55


Simple Queue

customers

Arrival process

Buffer

ServerDeparture of

Kendall notation

• Queue A/B/C/K/m/Z

• A: inter-arrival distribution

• B: service distribution

• C: number of servers

• K: buffer capacity (default ∞)

• m: customer population (default ∞)

• Z: buffer mechanism (default FIFO)

56


Simple Queue

customers

Arrival process

Buffer

ServerDeparture of

A and B distributions can be :

• M: exponential

• G: generic

• D: constant

• Ek: Erlang-k

• Hk: Hyperexponential-k

• Ck: Cox-k

• etc

57


Multi-class queues

• Customers of different classes can experience different arrival process, or service times...

Service distribution 2

Arrival process 1

Arrival process 2

Service distribution 1

58


Network of queues

• Several connected simple queues

• Open networks: customers can leave the system and are coming from the outside

p12

1

2

3p01

p02

p13

p23 p20

• Closed networks: finite population, customers can’t leave the system

3

1

2

59


Packet network

• Is modeled by a network of queues

• A network node: arrival buffer + processing time + sending buffers

• A transmission link: transmission time + propagation time

Transmission

...

...

Reception buffer

Processing

Sending buffer

Propagation

60


Performance parameters

Departures

Queuing System

XeQ

R

Xs

Arrivals

R1

tT0

N(t)

A1 D1A2 D2

R2

• Ak: arrival instant of the kth customer in the system

• Dk: departure instant of the kth customer in the system

• Rk = Dk −Ak: time spent in the system by the kth customer

• T (n, T ): time during which the system has n customers :∑∞n=0 T (n, T ) = T

• A(T )/D(T ): number of arrivals/departures in [0;T ]

61



• Input load: Xe(T ) = A(T )/T

• Throughput: Xs(T ) = D(T )/T

• Mean number of customers: Q(T ) = 1T

∑∞n=0 nT (n, T )

• Mean time spent in the system: R(T ) = 1A(T )

∑A(T )k=1 Rk

• Utilization ratio: U(T ) = 1− T (0, T )/T

62


Stability

• A system is stable iff the input load equals the throughput in the stationary regime

• i.e. limT→∞Xe(T ) = limT→∞Xs(T )

• or limT→∞D(T )A(T ) = 1

• A mono-class open networks of queues is stable iff all queues of the network are stable

• A finite population system is always stable

63


Little law

• For a stable system in stationary regime :

Q = RX

Qs=U=SX=Sl

l X

R

l

l

X

X

Qw

Q

WQs

S

Q=RX=Rl

Qw=WX=Wl

64


Distributions at arrival and departure instants

• A(n, T ): number of customers who find n customers in the system at their arrival

• D(n, T ): number of customers who leave n customers in the system at their departure

• P (n, T ) = T (n, T )/T : proportion of time during which there are n customers in thesystem

• Pa(n, T ) = A(n, T )/A(T ): proportion of clients that find n customers in the system

• Pd(n, T ) = D(n, T )/D(T ): proportion of clients that leave n customers in the system

• p(n) = limT→∞ P (n, T ): stationary probability to have n customers in the system

• pa(n) = limT→∞ Pa(n, T ): stationary probability for a client to find n customers

• pd(n) = limT→∞ Pd(n, T ): stationary probability for a client to leave n customers

65


Properties

• In a (stable and ergodic) system where customers arrive or leave one by one:

pa(n) = pd(n)

• In a system with Poisson arrivals:

pa(n) = p(n)

66

Simple Queues

1. M/M/1

(a) Definition

(b) Associated CTMC

(c) Stability

(d) Analysis

2. M/M/1/K

(a) Definition

(b) Associated CTMC

(c) Stability

(d) Analysis

3. Markovian queues

67

Simple Queues

M/M/1, Definition

• Infinite capacity

• A single server

• Poisson arrivals of rate λ

• Exponential service time of rate µ

68

Simple Queues

M/M/1, Associated DTMC

• Generic method: Find the ”right” state description → stochastic process → analysis →performance parameters

• What is a ”right” state description ? The knowledge of the present state is sufficient

to predict the evolution of the system = the generated stochastic process is a Markov

chain

• State description for the M/M/1: {n(t)}t≥0

69

Simple Queues

M/M/1, Associated DTMC

• Stochastic process with discrete state space and continuous time

• Memoryless property of the service time: it is not necessary to know for how long timethe service has started to predict the remaining service time

• Memoryless property of the inter-arrival law: it is not necessary to know when the lastcustomer arrived to predict the arrival of the next one

70

Simple Queues

M/M/1, Stability

• Stability condition:

λ < µ

ρ =λ

µ< 1

• ρ is referred to as the traffic intensity

71

Simple Queues

M/M/1, Analysis

• We assume that the system is stable

• Three methods to derive the stationary (or steady-state) probabilities:

• (1) Solve pQ = 0 and∑∞n=0 p(n) = 1

• (2) Solve the state equations: for each state, input flow=output flow

• (3) Solve the border equations

72

Simple Queues

M/M/1, Analysis

• pQ = 0

Q =

−λ λ 0 ... ... ...µ −(λ+ µ) λ 0 ... ...0 µ −(λ+ µ) λ 0 ...... 0 µ −(λ+ µ) λ 0... ... 0 µ −(λ+ µ) λ... ... ... 0 µ ...

73

Simple Queues

M/M/1, Analysis

• state equations: input flow=output flow

p(0)λ = p(1)µ

p(1)(λ+ µ) = p(0)λ+ p(2)µ

...

p(n)(λ+ µ) = p(n− 1)λ+ p(n+ 1)µ for all n ≥ 1

74

Simple Queues

• border equations:

p(0)λ = p(1)µ

p(1)λ = p(2)µ

...

p(n)λ = p(n+ 1)µ for all n ≥ 0

• p(n) = ρp(n− 1)

• p(n) = ρnp(0)

75

Simple Queues

• border equations:

p(0)λ = p(1)µ

p(1)λ = p(2)µ

...

p(n)λ = p(n+ 1)µ for all n ≥ 0

• Normalization : p(0) = 1∑∞n=0 ρ

n = 1− ρ

• Thus, for ρ < 1, ∀n ≥ 0 p(n) = (1− ρ)ρn

76

Simple Queues

M/M/1, Performance parameters

• Throughput:

X = Proba[queue non empty]µ =∞∑n=1

p(n)µ = ρµ = λ

• Server utilization:

U = 1− p(0) = ρ =λ

µ

• Average number of customers:

Q =∞∑n=1

np(n) =ρ

1− ρ

• Average time spent by a customer in the system (Little law):

R =Q

X=

1

µ(1− ρ)=

1

µ+

ρ

µ(1− ρ)

77

Simple Queues

M/M/1, Performance parameters

• Q and R:

78

Simple Queues

M/M/1/K, Definition

• The system (buffer and server) is limited to K customers

• A single server


• Exponential service time of rate µ

79

Simple Queues

M/M/1/K, Associated CTMC

• State description: {n(t)}t≥0

• Stochastic process with continuous time and discrete state space

• At a given time, when there is 0 < n(t) < K customers in the system, both inter-arrivaland service times are memoryless

80

Simple Queues

M/M/1/K, Stability

• The system is limited → the system is always stable

81

Simple Queues

M/M/1/K, Analysis

• Border equations:

∀n, 1 ≤ n ≤ K, p(n− 1)λ = p(n)µ

∀n, 1 ≤ n ≤ K, p(n) = ρp(n− 1)

∀n, 0 ≤ n ≤ K, p(n) = ρnp(0)

• Normalization:

p(0) =1∑K

n=0 ρn

=1− ρ

1− ρK+1

82

Simple Queues

M/M/1/K, Performance parameters

• Throughput X: two methods

• (1) Departure rate from the server:

Xs = Proba[queue not empty]µ = (1− p(0))µ =ρ− ρK+1

1− ρK+1µ

83

Simple Queues


• Throughput X: two methods

• (2) Arrivals in the system (not lost):

Xe = Proba[queue not full when a customer arrives]λ

=K−1∑n=0

pa(n)λ

=K−1∑n=0

p(n)λ

=1− ρK

1− ρK+1λ

84

Simple Queues


• Server utilization U :

U = 1− p(0) = ρ1− ρK

1− ρK+1

• Average number of customers Q:

Q =K∑n=0

np(n) =ρ

1− ρ×

1− (K + 1)ρK +KρK+1

1− ρK+1

85

Simple Queues


• Average time spent by a customers in the system: 2 possible parameters

• (1) Time spent by a customers admitted in the system: R = Q/X

• (2) Time spent by all customers trying to enter the system: RT = Q/λ

86

Simple Queues

Markovian queues, Definition

• Infinite buffer

• A single server

• Inter-arrival and service times law are memoryless and depends on the number of cus-tomers in the system

• When there are n customers in the system at a given time instant:

• - the arrival time of the next customer follows an exponential law of rate λ(n)

• - the service time follows an exponential law of rate µ(n)

87

Simple Queues

Markovian queues, Associated CTMC

• State description: {n(t)}t≥0


88

Simple Queues

Markovian queues, Stability


p(0) > 0

1

1 +∑∞n=1

(∏nk=1

λ(k−1)µ(k)

) > 0

∞∑n=1

n∏k=1

λ(k − 1)

µ(k)

<∞

89

Simple Queues

Markovian queues, Analysis

• Border equations:

∀n > 0, p(n− 1)λ(n− 1) = p(n)µ(n)

p(n) =n∏

k=1

λ(k − 1)

µ(k)p(0)

• Normalization:

p(0) =1

1 +∑∞n=1

(∏nk=1

λ(k−1)µ(k)

)

90

Simple Queues

Markovian queues, Performance parameters

• Throughput X: 2 methods

• (1) At the exit of the system:

X =∞∑n=1

p(n)µ(n)

• (2) At the entrance of the system:

X =∞∑n=0

pa(n)λ(n) =∞∑n=0

p(n)λ(n)

• Average number of customers in the system Q:

Q =∞∑n=1

np(n)

• Average time spent by a customer in the system R: R = Q/X

91

Erlang B and C laws

Telephonic traffic

N junctions/time−slots

Base−stationSwitch

• Main issue: find the right number of servers associated to a given quality of service(QoS)

• Traffic a of a server (ITU-T E-100) : average proportion of occupation time of theserver (t is the occupation time between 0 and T , T is the observation time) :

a =t

T

• a is in Erlangs

• US : Cent Call Second (CCS) is the occupation of a machine during 100s per hour

1CCS = 100/3600 = 1/36Erlangs

92

Erlang B and C laws

Telephonic traffic

M1T

...ML

• Traffic of a group of L servers:

A =

∑i tMi

T≤ LErlangs

• Identical servers, with n: mean number of occupation periods in T and τ : mean occu-pation time of a server:

A =nτ

T

• Ergodicity: statistics of the group at a given time and statistics of a given server in timeare equal

• with x: mean number of occupied servers at a given time and the ergodicity property:

x

L= a

93

Erlang B and C laws

Erlang B model, assumptions

• We consider S servers serving phone calls

• Calls arrive according to a Poisson law of rate λ, λ does not depend on the number of

occupied servers (infinite population assumption)

• Calls that cannot be served are lost

• Service time is distributed according to a decreasing exponential law of rate µ

94

Erlang B and C laws

Erlang B, markov chain

• The system is modeled with a M/M/S/S.

• Stationary probabilities are given by :

p(n) = ρn/n!p(0) and p(0) =

S∑k=0

ρk/k!

−1

with ρ = λ/µ

• Blocking probability is given by :

Pb = p(S) =ρS/S!∑Sk=0

ρk

k!

95

Erlang B and C laws

Erlang B law is not linear:

96

Erlang B and C laws

Erlang B law is not linear:

97

Erlang B and C laws

A recursive definition of the blocking probability:

Pb(S + 1) =ρPb(S)

S + 1 + ρPb(S).

An approximation of the inverse of the Erlang B law (Claude Rigault ENST) :

If E1,S(A) = ε = 10−k then S ≈ A+ k√A

• Example: consider a traffic of A = 100 Erlangs

• What is the number of junctions to install in order that the probability of lost call is less

than ε = 10−4?

• N ≈ 140 servers

98

Erlang B and C laws

Erlang C model, assumptions

• We consider S servers serving phone calls

• Calls arrive according to a Poisson law of rate λ, λ does not depend on the number of

occupied servers (infinite population assumption)

• Calls that cannot be served are put in an infinite buffer, discipline is FIFO

• Service time is distributed according to a decreasing exponential law of rate µ

99

Erlang B and C laws

Erlang C, markov chain

• The system is modeled with a M/M/S/∞.

• Stationary probabilities are given by :

p(n) = ρn/n!p(0) for 0 ≤ n ≤ S − 1,

p(n) = ρn/(S!Sn−S)p(0) for n ≥ S,

p(0) =

S−1∑k=0

ρk/k! +ρS/S!

1− ρ/S

−1

with ρ = λ/µ

100

Erlang B and C laws

Erlang C, waiting probability

Pw =∑n≥S

p(n)

=∑n≥S

ρS

S!

ρn−S

Sn−Sp(0)

=ρS

S!

S

S − ρp(0)

=

ρS

S!SS−ρ∑S−1

k=0 ρk/k! + ρS/S!

1−ρ/S

.

101

Erlang B and C laws

Erlang C, some useful results

• Mean number of customers:

Q = Pwρ/S

1− ρ/S

• Mean waiting time

W =Pw

λ

ρ/S

1− ρ/S

• Distribution of the waiting time:

P [T > t] = Pw exp(−Sµt(1− ρ/S))

102

Erlang B and C laws

Erlang C law is not linear:

103

Networks of Queues

1. Definition of the Jackson networks

2. Stability

3. Visit rates

4. Analysis and Jackson theorem

5. Performance parameters

104

Networks of Queues

Definition of the Jackson networks

• Open network

• M stations

• A single class of customers


• For each station: a single server, an unlimited FIFO buffer, an exponential service time

of rate µi

• Probabilistic routing: p0i=probability that a customers arriving in the network goes to

station i; pij=probability that a customer leaving station i goes to station j; pi0=probability

that a customer leaving station i goes out of the system

∀i, 0 ≤ i ≤M,M∑j=0

pij = 1

105

Networks of Queues

Example of Jackson network

106

Networks of Queues

Stability

• Let ei the average number of times that a customer visits station i during its journey in

the system

• ei is the visit rate of station i

• Arrival rate of customers in the network: λ

• Arrival rate of customers in station i: λi = eiλ


∀i = 1, ...,M, λi < µi

107

Networks of Queues

Visit rates

• The traffic λi at station i is made of :

• (1) the outside traffic: λp0i

• (2) the traffic coming from station j: λjpji for all stations j = 1, ...,M

λi = λp0i +M∑j=1

λjpji

• With λi = eiλ:

ei = p0i +M∑j=1

ejpji

108

Networks of Queues

Analysis

• State description: {n(t) = [n1(t), ..., nM(t)]}


• At a given instant, there is ni(t) customers in station i

• - arrivals are Poisson → memoryless

• - service times are exponential → memoryless

• - routing is probabilistic → memoryless

109

Networks of Queues

Jackson theorem

• For a stable Jackson network:

p(n) =M∏i=1

pi(ni)

• pi(ni) is the steady-state probability of a M/M/1 with arrival rate λi and service rate µi

• pi(ni) = (1− ρi)ρnii

• with ρi = λi/µi

110

Networks of Queues


• For station i:

Xi = λi Qi =ρi

1− ρiRi =

QiXi

=1

µi − λiUi = ρi

• For the network:

X = λ Q =M∑i=1

Qi

R =Q

X=Q

λ=

M∑i=1

Qiλ

=M∑i=1

eiQiλi

=M∑i=1

eiRi

111

introduction to network performance and queuing theory

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