International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 885
ISSN 2229-5518
Fuzzy Markov Model for Web server Queues
(1)Dr. T. Chitrakala Rani, (2)R. Neelambari
Abstract:
In this paper queuing network model which can be used for web traffic networks has been studied. The model is an open network of infinite server queues where client arrive according to a non- homogenous fuzzy Poisson process. The interaction of the client with server is described by a Fuzzy Markov Renewal Process. The changing attribute of the customer in the customer are driven by continuous time Markov chain and therefore change as they move through the network. The transient and limited number of customers in disjoint sets of servers and attributes are investigated. These turns out to be independent fuzzy Poisson random variable. The covariance of number of clients in 2 sets of servers and attributes at different time epochs has also been discussed.
Index Terms— Continuous time Markov chain (CTMC), independent fuzzy Poisson random variable, Non Homogeneous Fuzzy Poisson
Process(NHFPP), Fuzzy Markov Model, Fuzzy Markov renewal Process, Fuzzy Semi Markov kernel, Covariance.
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open network model of infinite server queues where client
In this paper traffic model for a web multimedia wireless network has been considered. Characterized by interaction between client server network and fuzzy web traffic aspects. We suppose that there exists a finite set of severs. This choice avoids the detailed location description of client interaction with server in order to obtain an integrated manageable model. Since in real time the arrival rate of call vary significantly over time we consider that clients arrive according to a non homogeneous fuzzy Poisson process. The successive server visited by a client form a discrete time Markov chain and the server residence time have a general distribution which depends on the call being visited as well as both the previous server visited and the next server to be entered.
A new method have been introduced for finding fuzzy system reliability using fuzzy profust reliability theory[2]. In [5] the theoretical aspects of modeling the uncertainty associated with links are discussed, whereas in [1] unified fuzzy Markov model has been used in a communication network .
Web is being used by all short of people for different purpose.
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• Dr. T. Chitrakala Rani , Associate Professor, Kundavai Naatchiyar
Governemnt Arts College for women(Autonomous), Thanjavur, India.
However with the aim of making our system more clear we present ours as a queuing network. Here we speak about
transition state in the network in according a Fuzzy Markov Renewal process and arrival pattern of the client NHFPP. We also take into account the characteristic of the client according to a CTMC.
The model consists of a fuzzy web system of infinite work stations. A client entering for ‘job n’ to kth station. Let D= I ∪M∪O denotes the set of nodes of the network where,
I - denotes the input node which will be visited by the client only when they enters the network
O - denotes the output node which will be visited by the client when he leaves the network, once he leaves the system he is supposed to reenter the network only through I
M – Denoted the intermediate nodes where the client may stay and has a freedom of moving different area’s in search of information he requires.
A fuzzy set is the generalization of the crisp sets. A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to a degree to which that individual is similar or compatible with the concept represented by the fuzzy set.
Let A� be a fuzzy set, the membership function [6]
µA� (x) ∈ [0,1] is evaluated for A� at x∈R where [0,1] denotes the
interval of real number from 0 to 1 including 0 and 1.Then the
fuzzy sets are the subsets of real number system.
Let Γ be the universe of discourse and ψ be the power set of Γ. Then the possibility measure σ is a mapping defined as follows [6]:
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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 886
ISSN 2229-5518
σ ∶ ψ → [0, 1]
such that the following properties holds
i. σ(∅) = 0 and σ(Γ) = 1
ii. σ(⋃ Ai ) = sup(σ(⋃ Ai ))
for every arbitrary collection Ai of ψ. The triplet (Γ,ψ, σ) is
called as possibility space.
Let N∗ = inf {n ≥ 1; Xn , t > 0} denotes the number of node
transition made by an arbitrary customer before leaving the
network.
TN ∗ denotes the time of departure so that XN ∗ is the outside
node reached by the customer at departure.
Y l(t) denotes the location node of the l th customer at t unit of
A fuzzy Markov model [6] is the model which has a
finite number of states Si, i = 1,2,3,..m at each transition
n = 1,2,3,…m together with fuzzy possibilities [2]
𝑝�𝑖𝑗 = σ[Xn+1 = Sj �Xn = Si].
time after arriving the network.
{Y l(t), t > 0}~{Y(t), t > 0}
Where {Y(t), t > 0}, being the minimal semi markov process
associated to {(Xn , Tn ) ,n≥0}defined by
X , T ≤ t ≤ T
∗ for all t>0 and iϵI, j ϵO.
Let E = {1,2,3,…m} be the state space and
(Γ,ψ, σ) be a possibility space. Then
We have
XN , t ≥ TN
Xn represents the state and the nth transition and
Tn represents the time of nth transition.
The process ( Xn , Tn ), n∈ N is called non homogenous fuzzy
Markov Renewal process if
p� ij (t) = σ[(Y(t) = j|Y(0) = i)] = min�p� ij (t), �rij (t)�
Suppose that each customer in the network has
associated an attributes that many change with time. Let Al(t)
σ[Xn = j, Tn ≤ t�X0 , X1 , … . Xn−1 = i, T0 , T1 , … Tn−1 = s]
denotes the attributes of lth
customers at t unit of time after
= σ[Xn = j, Tn ≤ t�X 〱−1 = i, Tn−1 = s] ; i ≠ j .
arriving the network[7].
Suppose if
Let Xn
denotes location node of the customer at T n
time.
{Al (t) , t > 0}~{A(t) , t > 0} with {Al ( t) , t > 0} being a
n n n n
continuous time markov chain with finite state space S.
{( Xl , Tl ), n ≥ 0}~{( X
where
, T ), n ≥ 0} --------------(1)
For t ≥ 0 and i, j ∈ S we have
l
Xn represents the steps at the n transition
q� i (t) = P[A (t) = j]
l l
Tn represents the time at the n transition
The process (Xn,Tn ) n∈ N is called non- homogeneous Fuzzy
Markov Renewal process if
q� ij (t) = P�{A (t) = j�A (0) = i}�
Let us also suppose that the attribute of the customer and
their locations in the network are independent (i.e)
l l
σ(Xn+1 = j, Tn+1 ≤ t|X0 , X1 , X2 , … . Xn = i, T0 , T1 , … . Tn)
= σ(Xn+1 = j, Tn+1 ≤ t|Xn = i); for i≠ j
The Transition distribution measure follows a homogenous
fuzzy semi markov kernel [2].
Q� ij = σ(Xn+1 = j, Tn+1 ≤ t|Xn = i)
�rij (t) = limn→∞ Q� ij (s, t) , i, j ∈ E, j ≠ i
r�ij (t) represent the possibility of the client making his next
transition to state j given that he entered state i at time t.
However before the entrance to j the process holds for a time s in state i.
For i,j∈E. Define �G�ij (t), t ≥ 0� as the fuzzy distribution
function of the time spent by the customer at the visiting
node i, given that its next visiting node is j [2].
{A (t); t ≥ 0, l ≥ 1}⊥ {Y (t), t > 0, l ≥ 1}
The arrival process of the customer to the network[7] Using time dependent coloring Poisson process we determine. Theorem 1:
Suppose that a customer enters to the network at time s is
placed in cell k with probability 𝑝�𝑘 (s), k ∈ K be finite and independent on other arrivals, For t≥0 and k ∈ K, let MK (t), t ≥ 0 denotes the number of customer placed in cell k unit time ‘t’. Then the point process {µK (t), t ≥ 0}, k ∈ K are
independent Non- homogenous fuzzy Poisson Process with
intensity measure µK (dt) = 𝑝�𝑘 (t). λ(t)dt, , t ≥ 0, k ∈ K.. For all t≥ 0 the random varibale {MK (t), k ∈ K} are independent and
t
Q� ij , if Q� ≠ 0
∴G�ij (t) = �r�ij(t)
MK (dt) = Poisson( ∫0 µK (ds) ), k ∈ K
where
G�ij (t) =
Q� ij
1, if Q� ij = 0
=σ(Tn+1 − Tn ≤ t|Xn+1 = j, Xn = i)if p ij >0.
Note:
Let us suppose �si , l ≥ 1� denotes the sequence of arrival time
to the network through input node I and let {Ni (t), t ≥ 0}
r�ij(t)
denotes the associated counting process.
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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 887
ISSN 2229-5518
Then by the theorem 1 with k = I and P� (s) = υ� (s) we
If {B
, m = 1,2,3, … . . M} be a partition of the st M× A then the
i i m
conclude that {Ni (t), t ≥ 0} i ∈I are independent non
homogenous fuzzy Poisson process with intensity
measure [4].
Λi(dt) = υ�i (t)λ(t) dt, i∈I and t≥ 0.
fuzzy random variables then the random variable
YB1 (t), YB2 (t), … … . . YBm (t) are independent and
t
YBm (t)~P �� � � q� β (t − s) . P� (t − s). Λ (ds)�
Let B⊆ I ×E, Yβ (t) denotes the number of customers whose
pair of location node and attribute at time t belongs to the set
B. Then the number of customer that are in node j with
attribute β at the time Y(j,β) (t).
For t≥ 0, the random variables Y(j,β) (t) j ∈D, β∈A are
independent and
t
Y(j,β) (t) ~Poisson[� � q� β (t − s) . P� (t − s). Λ (ds)
i∈I (j,β)ϵBm 0
for 1 ≤ m ≤ M and t ≥ 0
Thus it has been shown that the number of customers whose
pair of location node and attribute at time t belongs to each one of disjoint sets of pairs of a node and an attribute are independent fuzzy Poisson random variable.
Here the covariance of the number of customers with given
i∈I 0
ij i
(single or multiple) pairs of location node and attribute at different times is determined.
Consider t≥ 0 If the l th customer enters the network through
node I at the time s , 0≤ s≤ t , then it will be located in node j
and have attributeβ at time t if
Y l(t − s) = j and Al(t − s) = j is independent of other
For t, (t + h) ≥ 0, j1 , j2 ∈ M , β1 , β 2 ∈ A
Cov( Y(j1 ,β1) (t), Y(j2, ,β2 )(t+h)]
customers. Thus for a customer arriving at time s, 0≤s≤t , and having input node α, the probability of being located in
t q β
= ∑ ∫ 1
(t − s).q β1β 2
(h)
node j with attribute β at time t is only function of its arrival
time s and is equal to υ�ij (t) given below
υ�β (s) = �P�(Y(t − s) = j, A(t − s) = β)�Y(0) = i�
Proof:
i ∈ I 0 × P[(Y(t − s) = j1
, Y(t + h − s) = j
2 Y(0) = i )Λ i
(ds)]
= p� ij (t − s)q� β (t − s).
For i ∈I, j∈M and β∈A, we identify a mark ( i, j, β) customers.
Therefore using theorem 1 with k = I x M x A and
Let t, (t + h) ≥ 0, j1 , j2 ∈ M , β1 , β 2 ∈ A
X ( j ,β )( j , β ) (t, t + h) denotes number of customers that he
1 1 2 2
(i,j,β) (s) = �
υ� (s) υ�β (s), 0≤s≤t
is in the node
j1 with attribute
β1 at the time t and in the
0, otherwise
node j2 with attribute
β 2 at time
t + h . We conclude that
X ( j ,β )( j , β ) (t, t + h) is a fuzzy Poisson random variable
for (i, j,β) ∈K , we conclude that X( i,j,β) (t)are independent
1 1 2 2
fuzzy Poisson random variable with mean
E[X( i,j,β) (t)]=∫0 P�( i,j,β) (t)Λ(ds)
with mean.
E X( j1 , β 1 )( j 2 , β 2 )
(t , t + h)] =
=∫0 q� β (t − s) . P�
t − s). Λ (ds)
t q β
∑ ∫ 1
(t − s).
β 1β 2
(h)
ij ( i
Thus the number of customers that have a given attribute and
are located in a fixed node of the network at time t are independent fuzzy Poisson random variable.
i ∈ I 0 × [(Y(t − s) = j1 , Y(t + h − s) = j 2 Y(0) = i)Λ i (ds)]
P
Let {Bm , m = 1,2,3, … . . M} be a partition of the set M× A, the
set of all pairs of a node and an attribute. The number of
Cov[N1
+ N 2
, N1
+ N 3
] = E[ N1 ]
customers whose pairs of location node and attributed at time
t belongs to the set Bm is
∴ Cov[Y (t ),Y (t + h)]= E[X (t, t + h)]
Y( j ,β ) (t ) = X ( j ,β )( j , β ) (t, t + h) + Y( j ,β ) (t )
1 1 1 1 2 2 1 1
YBm (t) = � Y(j,β) (t), 1 ≤ m ≤ M , t ≥ 0.
− X ( j ,β )( j , β ) (t, t + h)
(j,β)ϵBm
1 1 2 2
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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 888
ISSN 2229-5518
Y( j ,β ) (t ) = X ( j ,β )( j , β ) (t, t + h) + Y( j ,β ) (t + h)
2 2 1 1 2 2 2 2
− X ( j ,β )( j , β ) (t, t + h)
1 1 2 2
Hence the three random variables in the second member of last two equations are independent fuzzy Poisson random variable.
We wish to thank Dr. Dhanam, Associate Professor, Govt arts college, Pudukkotai for her guidance in many ways for shaping this paper.
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The transient number of customer that have a given attributed and are located in a fixed node are shown to be independent Poisson random variable.
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