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Spectral Expansion Solution Methodology
for QBD-M Processes and Applications
in Future Internet Engineering
Tien Van Do1, Ram Chakka2,andJ´anos Sztrik3
1Department of Networked Systems and Services
Budapest University of Technology and Economics
Budapest, Hungary
2Meerut Institute of Engineering and Technology (MIET),
Meerut 250005, India
3Faculty of Informatics, University of Debrecen,
Egyetem t´er 1, Po.Box 12, 4010 Debrecen, Hungary
Abstract. Quasi Simultaneous-Multiple Births and Deaths (QBD-M)
Processes are used to model many of the traffic, service and related
problems in modern communication systems. Their importance is on the
increase due to the great strides that are taking place in telecommuni-
cation systems and networks. This paper presents the overview of the
Spectral Expansion (SE) for the steady state solution of QBD-M pro-
cesses and applications in future Internet engineering.
Keywords: QBD-M, Compound Poisson Process, Spectral Expansion.
1 Introduction
The concept of Quasi Birth-Death (QBD) processes, as a generalization of the
classical birth and death M/M/1 queues was first introduced by [1] and [2] in
the late sixties. The states of a QBD process are described by two dimensional
random variables called a phase and a level [3–5] and transitions in a QBD pro-
cess are only possible between adjacent levels. It is observed that QBD processes
create a useful framework for the performability analysis of many problems in
telecommunications and computer networks [6–11].
In the QBD process, if the nonzero jumps in levels are not accompanied with
changes in a phase, then these processes are known as Markov-modulated Birth
and Death processes . The infinite number of states involved makes the solution
of these models nontrivial. There are several methods of solving these models,
either the whole class of models or any of the subclasses.
Seelen has analysed a Ph/Ph/c queue in this frame work [12]. Seelen’s method
is an approximate one, the Markov chain is first truncated to a finite state which
is an approximation of the original process. The resulting finite state Markov
chain is then analysed, by exploiting the structure in devising an efficient itera-
tive solution algorithm. The second method is to reduce the infinite state prob-
lem to a linear equation involving vector generating function and some unknown
N.T. Nguyen, T. Van Do, and H.A. Le Thi (Eds.): ICCSAMA 2013, SCI 479, pp. 131–142.
DOI: 10.1007/978-3-319-00293-4_11 c
Springer International Publishing Switzerland 2013
132 T. Van Do, R. Chakka, and J. Sztrik
probabilities. The latter are then determined with the aid of the singularities of
the coefficient matrix. A comprehensive treatment of that approach, in the con-
text of a discrete time process with a general M/G/1 type structure, is presented
in [13]. The third way of solving these models is the well known matrix-geometric
method, first proposed by Evans [2, 3]. In this method a nonlinear matrix equa-
tion is first formed from the system parameters and the minimal nonnegative
solution Rof this equation is computed by an iterative method. The invariant
vector is then expressed in terms of the powers of R. Neuts claims this method
has probablistic interpretation for the steps in computation. That is certainly
an advantage. Yet, this method suffers from the fact that there is no way of
knowing how many iterations are needed to compute Rto a given accuracy. It
can also be shown that for certain parameter values the computation require-
ments are uncertain and formidably large. The fourth method is known as the
spectral expansion method. It is based on expressing the invariant vector of the
process in terms of eigenvalues and left eigenvectors of a certain matrix polyno-
mial. The generating function and the spectral expansion methods are closely
related. However, the latter produces steady state probabilities directly using an
algebraic expansion while the former provides them through a transform.
It is confirmed by a number of works that the spectral expansion method is
better than the matrix geometric one from some aspects [4, 14, 15]. This paper
gives the overview of the SE methodology and explains how the SE methodology
is used towards the analysis of QBD-M processes and the performance evaluation
of ICT systems and future Internet.
The rest of the paper is organized as follows. In Section 2, the terminology
and definitions are presented. The spectral expansion methodology is provided in
Section 3. Examples are given in Section 4. The paper is concluded in Section 5.
2 Definitions
Consider a two-dimensional continuous time, irreducible Markov chain
X={(I(t), J(t)), t≥0}on a lattice strip.
–I(t) is called the phase (e.g., the state of the environment) of the system
at time t. Random variable I(t) takes values from the set {0,1,2,...,N},
where Nis the maximum value of the phase variable.
–Random variable J(t) is often called the level of the system at time t and
takes a set of values {0,1,...,L},whereLcan be finite or infinite.
The state space of the Markov chain Xis {(i, j):0≤i≤N, 0≤j≤L}.Let
pi,j denote the steady state probability of the state (i, j)as
pi,j = lim
t→∞ Pr(I(t)=i, J(t)=j); (i=0,...,N;j=0,1,...,L).
Vector vjis defined as
vj=(p0,j ,...,p
N,j)(j=0,1,...,L).
Spectral Expansion Solution Methodology 133
Since the sum of all the probabilities pi,j is 1.0, we have the normalization
equation as
L
j=0
vjeN+1 =1,(1)
where eN+1 is a column vector of size N+1 withall ones.
2.1 Continuous Time QBD Processes
Definition 1. A continuous time Quasi-Birth-and-Death (QBD) process is
formed when one-step transitions of the Markov chain Xare allowed to states in
the same level or in the two adjacent levels. That is, the dynamics of the process
are driven by
(a) purely phase transitions. Aj(i, k)denotes the transition rate from state (i, j )
to state (k, j)(0≤i, k ≤N;i=k;j=0,1,...,L);
(b) one−step upward transitions. Bj(i, k)is the transition rate from state (i, j)
to state (k, j +1) (0≤i, k ≤N;j=0,1,...,L);
(c) one−step downward transitions. Cj(i, k)is the transition rate from state
(i, j)to state (k, j −1) (0 ≤i, k ≤N;j=0,1,...,L).
Let Aj,Bjand Cjdenote (N+1)×(N+ 1) matrices with elements Aj(i, k),
Bj(i, k)andCj(i, k ), respectively. Note that the diagonal elements of matrix A
are zero. Let DAj,DBjand DCjbe the diagonal matrices of size (N+1)×(N+1),
defined by the ith (i=0,...,N) diagonal element as follows
DAj(i, i)=
N
k=0
Aj(i, k); DBj(i, i)=
N
k=0
Bj(i, k); DCj(i, i)=
N
k=0
Cj(i, k).
For the convenience of the presentation we define matrices B−1=0,BL=0
and C0=0.
The steady state balance equations satisfied by the vectors vjare
vj[DAj+DBj+DCj]=vj−1Bj−1+vjAj+vj+1Cj+1 ∀j. (2)
Assume that there exist thresholds T∗
1and T∗
2such that
Aj=A(T∗
2≥j≥T∗
1),
Bj=B(T∗
2≥j≥T∗
1−1),
Cj=C(T∗
2+1≥j≥T∗
1).
DA,DBand DCare the corresponding diagonal matrices with the diagonal
elements as
DA(i, i)=
N
k=1
A(i, k),D
B(i, i)=
N
k=1
B(i, k),D
C(i, i)=
N
k=1
C(i, k).
134 T. Van Do, R. Chakka, and J. Sztrik
The generator matrix of the QBD process is written as
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A(1)
0B000 ... ... ... ... ...
C1A(1)
1B10... ... ... ... ...
0C2A(1)
2B2... ... ... ... ...
.
.
..
.
..
.
....... ... ... ...
00... C
T∗
1
−1A(1)
T∗
1
−1BT∗
1
−100...
00... 0CT∗
1A(1)
T∗
1BT∗
10...
00... 00CT∗
1+1 A(1)
T∗
1+1 BT∗
1+1 ...
.
.
..
.
..
.
..
.
.... ... ...... ...
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A(1)
0B000 ... ... ... ... ... ...
C1A(1)
1B10... ... ... ... ... ...
0C2A(1)
2B2... ... ... ... ... ...
.
.
..
.
..
.
....... ... ... ... ... ...
00... C
T∗
1
−1A(1)
T∗
1
−1Q000... ...
00... 0CT1Q1Q00... ...
00... 00Q2Q1Q0... ...
00... 000Q2Q1Q0...
.
.
..
.
..
.
..
.
.... ... ...... ... ...
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
where A(1)
j=Aj−DAj−DBj−DCj.
The j-independent balance equations can be rewritten as follows
vj−1Q0+vjQ1+vj+1Q2=0 (T∗
1≤j≤T∗
2),(3)
where Q0=B, Q1=A−DA−DB−DC,Q
2=C.
2.2 Continuous Time QBD-M Processes
Definition 2. The Markov chain Xis called a continuous time quasi
simultaneous-bounded-multiple births and simultaneous-bounded-multiple deaths
(QBD-M) process if the balance equation for level jcanbewrittenas
y
i=0
vj−y1+iQi=0 (T1≤j≤T2),(4)
where y,y1,T1and T2are integer constants for a specific system, while Qiare
j-independent matrices of size (N+1)×(N+1).
2.3 Generalized Exponential Distribution
Definition 3. The versatile Generalized Exponential (GE) distribution is given
in the following form:
F(t)=P(W≤t)=1−(1 −φ)e−μt (t≥0),(5)
where Wis the GE random variable with parameters μ, φ.
Spectral Expansion Solution Methodology 135
Thus, the GE parameter estimation can be by obtained by 1/ν, the mean, and
C2
coeff , the squared coefficient of variation of the inter-event time of the sample
as
1−φ=2/(C2
coeff +1) ; μ=ν(1 −φ).(6)
Remarks.ForC2
coeff >1, the GE model is a mixed-type probability distribu-
tion having the same mean and coefficient of variation, and with one of the two
phases having zero service time, or a bulk type distribution with an underlying
counting process equivalent to a Batch (or Bulk) Poisson Process (BPP) with
batch-arrival rate μand geometrically distributed batch size with mean 1/(1−φ)
and SCV (C2
coeff −1)/(1 + C2
coeff ) (see [16]). It can be observed that there is
an infinite family of BPP’s with the same GE-type inter-event time distribu-
tion. It is shown that, among them, the BPP with geometrically distributed
bulk sizes (referred as the CPP) is the only one that constitutes a renewal pro-
cess (the zero inter-event times within a bulk/batch are independent if the bulk
size distribution is geometric [17]). The GE distribution is versatile, possessing
pseudo-memoryless properties which make the solution of many GE-type queu-
ing systems analytically tractable [17]. The choice of the GE distribution is often
motivated by the fact that measurements of actual inter-arrival or service times
may be generally limited and so only a few parameters (for example the mean
and variance) can be computed reliably. Typically, when only the mean and
variance can be relied upon, a choice of a distribution which implies least bias
is that of GE-type distribution [16, 17].
Definition 4 (CPP). The inter-arrival time distribution of customers of the
Compound Poisson Process (CPP) is GE with parameters (σ, θ). That is, the
inter-arrival time probability distribution function is 1−(1 −θ)e−σt.
Thus, the arrival point-process has batches arriving at each point having inde-
pendent and geometric batch-size distribution. Specifically the probability that
a batch is of size sis (1 −θ)θs−1.
3 The Spectral Expansion Method for QBD-M Processes
Let Q(λ) denote the characteristic matrix polynomial associated with the bal-
ance equation (4) as
Q(λ)=
y
i=0
Qiλi.(7)
If (λ, ψ) is the eigenvalue and left-eigenvector pair of the characteristic matrix-
polynomial, the following equation holds
ψQ(λ)=0; det[Q(λ)] = 0.(8)
Assume that Q(λ)hasdpairs of eigenvalue-eigenvectors. For the kth (k=
1,...,d) non-zero eigenvalue-eigenvector pair, (λk,ψk), by substituting vj=
ψkλj
k(T1−y1≤j≤T2−y1+y) in the equations (4), it can be seen that this
136 T. Van Do, R. Chakka, and J. Sztrik
set of equations is satisfied. Hence, that is a particular solution. The equations
can even be satisfied with ψkλj+lk
kfor any real lk. It is easy to prove that the
general solution for vjis the linear sum of all the factors (ψkλj−T1+y1
k)as
vj=
d
l=1
alψlλj−T1+y1
l(j=T1−y1,T
1−y1+1,...,T
2−y1+y),(9)
where al(l=1,...,d)areconstants.
Therefore, the steady state probability can be written as follows
pi,j =
d
l=1
alψl(i)λj−T1+y1
l(j=T1−y1,T
1−y1+1,...,T
2−y1+y).(10)
An interesting property can be observed concerning the eigenvalues of Q(λ)for
QBD-M process Xas follows. If (λk,ψk) is the left-eigenvalue and eigenvec-
tor pair of Q(λ), then (1/λk,ψk) is the left-eigenvalue and eigenvector pair of
Q(λ)=
y
i=0
Qy−iλi, the characteristic matrix polynomial of the dual process of
X(see [14]).
3.1 Infinite QBD-M Processes
When Land T2are infinite (unbounded), consider the probability sum
∞
j=T1−y1
pi,j =
∞
j=T1−y1
d
l=1
alψl(i)λj−T1+y1
l.(11)
In order to ensure that this sum is less or equal to 1.0, the necessary condition
is
ak=0,if|λk|≥1.
Thus, by renumbering the eigenvalues inside the unit circle, the general solution
is obtained as
vj=
χ
l=1
alψlλj−T1+y1
l(j=T1−y1,T
1−y1+1,...),(12)
pi,j =
χ
l=1
alψl(i)λj−T1+y1
l(j=T1−y1,T
1−y1+1,...).(13)
where χis the number of eigenvalues that are present strictly within the unit
circle. These eigenvalues appear some as real and others as complex-conjugate
pairs, and as do the corresponding eigenvectors.
In order to determine the steady state probabilities, the unknown constants
alare to be determined. Their number is χ. We still have other unknowns
Spectral Expansion Solution Methodology 137
v0,v1,...,vT1−y1−1. These unknowns are determined with the aid of the state
dependent balance equations (their number is T1(N+ 1)) and the normalization
equation (1), out of which T1(N+ 1) are linearly independent. These equations
can have a unique solution if and only if (T1−y1)(N+1)+χ=T1(N+1), or
equivalently
χ=y1(N+ 1) (14)
holds.
3.2 Finite QBD-M Processes
In order to compute the steady state probabilities, the unknown constants
alare to be determined. Their number is d. We still have other unknowns
v0,v1,...,vT1−y1−1,vT2−y1+y+1,vT2−y1+y+2,...,vL. Therefore, the number of
unknowns is
d+(T1−y1)(N+1)+(L−T2+y1−y)(N+1).
These unknowns are determined with the aid of the state dependent balance
equations (their number is T1(N+1)+(L−T2)(N+ 1)) and the normalization
equation, out of which T1(N+1)+(L−T2)(N+ 1) are linearly independent.
These equations can have a unique solution if and only if
d+(T1−y1)(N+1)+(L−T2+y1−y)(N+1)=T1(N+1)+(L−T2)(N+1),
equivalently
d=y(N+ 1) (15)
holds.
4 Examples and Applications
Example 1 (M/M/c/L queue with breakdowns and repairs). The queue with an
infinite buffer is described by the Markov chain {I(t),J(t)},whereI(t)-the
operative state of the system- represents the number of operative servers at
time tand J(t) is the number of jobs in the system at time t, including those
being served. The maximum number of operative servers is c.TheMarkovchain
is irreducible with state space {0,1,...,c}×{0,1,...,L}. Note that in this
example the phase is numbered from 0 and the transition rate matrices are of
size (c+1)×(c+1). The number of phases is N+1 = c+ 1. Jobs arrive according
to an independent Poisson process with rate σ. The service rate of an operative
server is denoted by μ. Processors break down independently at rate ξand are
repaired at rate η. When a new job arrives or when a completed job departs
from the system, the operative state does not change.
138 T. Van Do, R. Chakka, and J. Sztrik
The matrices Ajand Aare given by
A=Aj=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0cη
ξ0(c−1)η
2ξ0...
......η
cξ 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(j=0,1,...).(16)
The one-step upward transitions are created by the arrivals of single jobs. There-
fore, Band Bjthe one-step upward transition rate matrices are
B=Bj=diag[σ,σ,...,σ](j=0,1,...).(17)
The one-step downward transitions take place by the departures of single jobs,
after their service completion. The departure rate (Cj(i, i)) of jobs at time t
depends on I(t)=iand J(t)=j.Ifi>j, then a server is assigned to every job
and not all operative servers are occupied, hence the departure rate Cj(i, i)=
jμ.Ifi≤j, then all the operative processors are occupied by jobs, hence the
departure rate Cj(i, i)=iμ.NotethatCj(i, i) does not depend on jif j≥i.
Therefor, Cjdoes not depend on jif j≥c.
Cj=diag[0,min(j, 1)μ, min(j, 2)μ,...,min(j, c)μ](0<j<c),
C=diag[0,μ,2μ,...,cμ](j≥c),
C0=0.(18)
The M/M/c/L queue with breakdowns and repairs is an example of the QBD
process, where the coefficient matrices of the characteristic matrix polynomial
are Q0=B=B=diag[σ,σ,...,σ], Q1=A−DA−DB−DC,Q2=C.
Example 2 (Retrial queues to model DHCP [18]). The size of the pool (i.e.: the
number of allocatable IP addresses) is c. The fix lease time value sent by the
DHCP server is denoted by Tl. The inter-arrival times of DHCP requests are
exponentially distributed with a mean inter-arrival time 1/λ.
Assume that the holding times (i.e.: how long does a client need an IP address)
of clients are represented by random variable Hwith a cumulative distribution
function Pr(H<x)=F(x). Upon the expiration of the lease time, the previ-
ously allocated address at the DHCP server becomes free and can be allocated
to another client unless the client extends the use of a specific IP address before
the expiration of the lease time. Let adenote the probability that DHCP clients
leave (i.e.: switch off the computer) the system or do not renew the allocated IP
address after the expiration of its lease time. We can write
a=Pr(H<T
l)=F(Tl).
Let I(t) denote the number of allocated IP addresses at time t.Notethat0≤
I(t)≤cholds. A client who does not receive the allocation of an IP address
Spectral Expansion Solution Methodology 139
because the shortage (when I(t)=c)ofIPaddressessetsatimertowaitfora
limited time and will retry the request for an IP address upon the expiration of
backoff time. We model this phenomenon as the client joins the “virtual orbit”.
J(t) represents the number of DHCP clients in the ”orbit” at time tand takes
values from 0 to ∞.
Lease times are exponentially distributed with a mean lease time 1/μ =Tl.
Clients waiting in the orbit repeat the request for the DHCP server with rate ν
(i.e.: the inter-repetition times are exponentially distributed with parameter ν),
which is independent of the number of waiting clients in the orbit.
The evolution of the system is driven by the following transitions.
(a) Aj(i, k) denotes a transition rate from state (i, j) to state (k, j )(0≤i, k ≤
c;j=0,1,...), which is caused by either the arrival of DHCPDISCOVERY
requests or by the expiration of the lease time without the renewal of an allo-
cated IP address. Matrix Ajis defined as the matrix with elements Aj(i, k).
Since Ajis j-independent, it can be written as
Aj=A=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0λ0...000
aμ 0λ...000
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
00 ...a(c−1)μ0λ
00 ...0acμ 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
∀j≥0;
(b) Bj(i, k) represents one step upward transition from state (i, j ) to state
(k, j +1) (0 ≤i, k ≤c;j=0,1,...), which is due to the arrival of DHCPDIS-
COVERY requests when no free IP address is available in the IP address
pool. In the similar way, matrix Bj(B) with elements Bj(i, k) is defined as
Bj=B=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
000...000
000...000
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
00 ...000
00 ...00λ
⎤
⎥
⎥
⎥
⎥
⎥
⎦
∀j≥0;
(c) Cj(i, k) is the transition rate from state (i, j) to state (k, j −1) (0 ≤i, k ≤
c;j=1,...), which is due to the successful retrial of a request from the orbit.
Matrix Cj(∀j≥1) with elements Cj(i, k) is written as
Cj=C=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0ν0...000
00ν ... 000
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
00 ...00ν
00 ...000
⎤
⎥
⎥
⎥
⎥
⎥
⎦
∀j≥1.
140 T. Van Do, R. Chakka, and J. Sztrik
The infinitesimal generator matrix of Ycan be written as follows
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A00 B 0 ... ... ... ...
CQ1B 0 ... ... ...
0CQ1B 0 ... ...
00CQ1B 0 ...
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
... ... ... ... ... ... ...
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(19)
where DAand DCare diagonal matrices whose diagonal elements are the sum
of the elements in the corresponding row of Aand C, respectively. Note that
A00 =A−DA−B,Q1=A−DA−B−DC.
5 Conclusions
We have presented an overview for the spectral expansion method to solve QBD-
M processes which can be applied to evaluate the performance of various systems,
services in information and communication technology (ICT)systems and future
Internet. The spectral expansion method is proved to be a mature technique for
the performance analysis of various problems [4, 6, 7, 14, 19–38]. The examples
include the performance evaluation of Optical Burst/Packet (OBS) Switching
networks [24, 39], MPLS networks [23, 30], the Apache web server [7], and wire-
less networks [6, 24, 28, 40].
Acknowledgement. The publication was supported by the T´
AMOP-4.2.2.C-
11/1/KONV-2012-0001 pro ject. The project has been supported by the Euro-
pean Union, co-financed by the European Social Fund.
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