A Generalized Markovian Queue and Its Applications to Performance Analysis in Telecommunications Networks Ram Chakka a,1, Tien Van Do b,∗, Zsolt P´andi b a
Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam 515 134, India b Department
of Telecommunications, Budapest University of Technology and Economics, Magyar tud´ osok k¨ or´ utja 2., Budapest, H-1117, Hungary
Abstract P In this paper the MM K k=1 CP Pk /GE/c/L G-queue is introduced and proposed as a generalised Markovian node model in telecommunications networks. An exact and computationally efficient solution is obtained for the steady-state probabilities and performance measures. Issues concerning the computational effort are also discussed. The proposed queue is applied to the performance analysis of optical burst switching (OBS) nodes. The numerical results obtained and also the numerical results of a previous model are compared to the simulation results of the OBS obtained using captured traffic traces. We have also introduced negative customers into the model, in an innovative way, in order to account for the loss of packets due to technology limitations of the FDL’s (fiber delay loop), which is rather specific to the optical domain. The model is quite promising as a viable performance predictor. Key words: generalized Markovian queue, Optical Burst Switching
∗ Corresponding author Email addresses:
[email protected] (Ram Chakka),
[email protected] (Tien Van Do),
[email protected] (Zsolt P´andi). 1 Ram Chakka is presently a honorary visiting faculty with the Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam 515 134, India, on leave from the Department of Computer Science, Norfolk State University, Norfolk, VA 23504, USA.
Preprint submitted to Elsevier Science
8 June 2004
1
Introduction
A number of models are being developed for the traffic, service and queuing in the emerging telecommunication networks. The task is rather difficult because the traffic is bursty and correlated. These models include the compound Poisson process (CPP) [1], the Markov modulated Poisson process (MMPP) and self-similar traffic models such as the Fractional Brownian Motion (FBM) [2]. P The CPP and the K k=1 CP Pk traffic models often give a good representation of the burstiness (batch size distribution) of the traffic from one or more sources [3], but not the auto-correlations of the inter-arrival time observed in real traffic. Conversely, the MMPP models can capture the auto-correlations but not the burstiness [4,5]. Self-similar traffic models such as the FBM can represent both burstiness and auto-correlations, but they are analytically intractable in a queuing context. The traffic arriving to a node is often the superposition of traffic from a number of sources (homogeneous or heterogeneous), which further complicates the analysis of the system. In order to model such systems in the Markovian framework, a new trafP fic/queuing model, the Markov modulated K k=1 CP Pk /GE/c/L G-queue is introduced in this paper. This is a homogeneous multi-server queue with c servers, GE (generalized exponential) service times. The arrivals are the superposition of K independent positive customer (referred as, customers) and an independent negative customer streams 2 . Each of these arrival streams is a CPP, i.e. a Poisson point process with batch arrivals of geometrically distributed batch size. Furthermore, all the K + 2 independent GE distributions (K positive and one negative customer inter-arrival times and also the service time) are jointly modulated by a continuous time Markov phase process P (CTMP), also termed as the modulating process. This MM K k=1 CP Pk /GE/c/L G-queue and its extensions can capture a large class of traffic and queuing models applicable to today’s Internet in the Markovian framework. Our contributions are both theoretical and practical. On the theoretical side, P we propose the MM K k=1 CP Pk /GE/c/L G-Queue and present a methodology for the steady state solution of the new queue. The methodology includes a series of transformations of the balance equations in order to reach a computable form, that of the QBD-M (Quasi Simultaneous-Bounded-Multiple Births and simultaneous-Bounded-Multiple Deaths) type. As a consequence, existing methods (either the spectral expansion method [8] or an appropriate extension of Naoumov’s method [9] for QBD processes based on the matrixgeometric solution approach [10]) can be employed to obtain the steady state 2
Negative customers remove (positive) customers from the queue and have been used to model random neural networks, task termination in speculative parallelism, faulty components in manufacturing systems and server breakdowns [6].
2
probabilities. On the practical side, we illustrate the utility of the new queue by the performance analysis of an Optical Burst Switching (OBS) 3 multiplexer and storage. Turner [11] has proposed a birth-death process to analyze this system, however, that model has certain limitations such as the assumption of an exponential inter-burst arrival time and constant burst size. Our model overcomes all those limitations, hence it is much more accurate to incorporate the properties of real traffic. This is also confirmed by comparing with the results obtained by trace driven simulations of the OBS using captured traffic traces from the Internet Traffic Archive. Negative customers are also applied in the model in a novel and innovative way to account for packet losses in the OBS specifically due to certain technology limitations of a storage solution using the FDL’s (fiber delay loop). The rest of the paper is organized as follows. The MM
K X
CP Pk /GE/c/L G-
k=1
queue is described in Section 2. In Section 3, a solution technique is presented for obtaining the exact steady state probability distribution of the Markov process that represents the system. An extensive, application oriented, numerical study is carried out in Section 4, and the paper concludes in Section 5.
2
The MM
K X
CP Pk /GE/c/L G-queue
k=1
2.1 The modulating process
The arrival and service processes are modulated by the same continuous time, irreducible Markov phase process with N states. Let Q be the generator matrix of this process, given by
3
Packet and burst switching have been proposed for optical networks because they can better accommodate bursty traffic generated by IP applications. OBS networks switch bursts of variable length while the payload data stays in the optical domain[11–13].
3
−q1 q1,2 . . . q1,N q2,1 −q2 . . . q2,N , Q= . .. . . .. .. . . .
qN,1 qN,2 . . . −qN
where qi,k (i 6= k) is the instantaneous transition rate from phase i to phase k, P and qi = N (i = 1, . . . , N ). Let r = (r1 , r2 , . . . , rN ) be the vector j=1 qi,j , of equilibrium probabilities of the modulating phases. Then, r is uniquely determined by the equations, rQ = 0 ; reN = 1, where eN stands for the column vector with N elements, each of which is unity.
2.2 The customer arrival process The arrival process, in any given modulating phase, is the superposition of K independent CPP arrival streams of customers (or packets, in packet-switched networks) and an independent CPP of negative customers. Customers of different arrival streams are not distinguishable. The parameters of the GE interarrival time distribution of the k th (1 ≤ k ≤ K) customer arrival stream in phase i are (σi,k , θi,k ), and (ρi , δi ) are those of the negative customers. That is, the inter-arrival time probability distribution function is 1 − (1 − θi,k )e−σi,k t , in phase i, for the k th stream of customers and 1 − (1 − δi )e−ρi t for the negative customers. Thus, all the K + 1 arrival point-processes are Poisson, with batches arriving at each point having geometric size distribution. Specifically, s−1 in phase i, the probability that a batch is of size s is (1 − θi,k )θi,k for the k th stream of customers, and (1 − δi )δis−1 for the negative customers. Let σi,. , σi,. be the average arrival rate of customer batches and customers in phase i respectively. Let σ, σ be the overall average arrival rate of batches and customers respectively. Then, σi,. =
K X
k=1
σi,k ,
σi,. =
K X
σi,k , k=1 (1 − θi,k )
σ=
N X
σi,. ri ,
σ=
i=1
N X
σi,. ri (1)
i=1
2.2.1 Arrival batch size distribution of positive customers The overall arrival stream of customers is thus an MM K k=1 CP Pk . Because of the superposition of many CPP’s, the overall arrivals in phase i can be considered as bulk-Poisson (M [x] ) with arrival rate σi,. and with a batch size distribution {πi,l }, that is more general than mere geometric. The probability P
4
that the batch size is l (πi,l ) and the overall batch size distribution (π.,l ), can be given by πi,l =
N X σi,k l−1 (1 − θi,k )θi,k , π.,l = ri πi,l i=1 k=1 σi,. K X
(2)
Clearly, by choosing K and other parameters appropriately it may be possible to approximate {πi,l } or {π.,l } to suit certain given classes of batch size distribution, however this is only an objective of further research.
2.3 The GE multi-server
The service facility has c homogeneous servers in parallel, each with GEdistributed service times with parameters (µi , φi ) in phase i. The service discipline is FCFS and each server serves at most one positive customer at any given time. Negative customers neither wait in the queue, nor are served. The operation of the GE server is similar to that described for the CPP arrival processes above. L denotes the queuing capacity, in all phases, including the customers in service. L can be finite or infinite. We assume, when the number of customers is j and the arriving batch size of positive customers is greater than L − j (assuming finite L), that only L − j customers are taken in and the rest are rejected. However, the batch size associated with a service completion is bounded by one more than the number of customers waiting to commence service at the departure instant. For queues of length c ≤ j < L+1 (including any customers in service), the maximum batch size at a departure instant is j − c + 1, only one server being able to complete a service period at any one instant under the assumption of exponentially distributed batch-service times. Thus, the probability that a departing batch has size s is (1 − φi )φs−1 for 1 ≤ s ≤ j − c i j−c and φi for s = j − c + 1. In particular, when j = c, the departing batch has size 1 with probability one, and this is also the case for all 1 ≤ j ≤ c since each customer is already engaged by a server and there are then no customers waiting to commence service. It is assumed that the first positive customer in a batch arriving at an instant when the queue length is less than c (so that at least one server is free) never skips service, i.e. always has an exponentially distributed service time. However, even without this assumption the methodology described in this paper is still applicable. 5
2.4 Negative customer semantics A negative customer removes a positive customer in the queue, according to a specified killing discipline. We consider here a variant of the RCE killing discipline (removal of customers from the end of the queue), where the most recent positive arrival is removed, but which does not allow a customer actually in service to be removed: a negative customer that arrives when there are no positive customers waiting to start service has no effect. We may say that customers in service are immune to negative customers or that the service itself is immune servicing. Such a killing discipline is suitable for modeling e.g. load balancing, where work is transferred from overloaded queues but never work, that is, actually in progress. When a batch of negative customers of size l (1 ≤ l < j − c) arrives, l positive customers are removed from the end of the queue leaving the remaining j − l positive customers in the system. If l ≥ j −c ≥ 1, then j −c positive customers are removed, leaving none waiting to commence service (queue length equal to c). If j ≤ c, the negative arrivals have no effect. ρi , the average arrival rate of negative customers in phase i and ρ, the overall average arrival rate of negative customers are given by, ρi =
ρi ; 1 − δi
ρ=
N X
ri ρ i
(3)
i=1
2.5 Condition for stability When L is finite, the system is ergodic since the representing Markov process is irreducible. Otherwise, i.e. when L = ∞, the overall average departure rate increases with the queue length, and its maximum (the overall average departure rate when the queue length tends to ∞) can be determined as, µ=c
N X i=1
r i µi . 1 − φi
(4)
Hence, we conjecture that the necessary and sufficient condition for the existence of steady state probabilities is, σ FK,l (Φ + ∆)
l=1
"
+ +
K X
#
(−1)l < j + 2 − l >FK,l Φ∆
l=1
(A.2)
Expanding and grouping the terms together, equation (A.2) can be written as,
(2)
←−
K X
< j − l >GK,l
(A.3)
l=−2
where, GK,l = (−1)l [FK,l + (Φ + ∆)FK,l+1 + Φ∆FK,l+2 ] (l = −2, −1, . . . , K) (A.4) The balance equations < j + 2 >, < j + 1 >, < j >, . . ., < j − l >, . . ., < j − K >, respectively are given by, 21
j+2 K XX
vj+2−s Θs−1 k (E − Θk )Σk + vj+2 [Q − Σ − Cj+2 − R]
s=1 k=1
+
L−j−2 X
vj+2+s Cj+2+s,j+2 = 0 ;
s=1
j+1 K XX
vj+1−s Θs−1 k (E − Θk )Σk + vj+1 [Q − Σ − Cj+1 − R]
s=1 k=1
+
L−j−1 X
vj+1+s Cj+1+s,j+1 = 0 ;
s=1
j X K X
vj−s Θs−1 k (E − Θk )Σk + vj [Q − Σ − Cj − R]
s=1 k=1
+
L−j X
vj+s Cj+s,j = 0 ;
s=1
.. . j−l X K X
vj−l−sΘs−1 k (E − Θk )Σk + vj−l [Q − Σ − Cj−l − R]
s=1 k=1
+
L−j+l X
vj−l+s Cj−l+s,j−l = 0 ;
s=1
.. . j−K K X X
vj−K−s Θs−1 k (E − Θk )Σk + vj−K [Q − Σ − Cj−K − R]
s=1 k=1
+
L−j+K X
vj−K+sCj−K+s,j−K = 0 ;
s=1
Substituting or applying the above to (A.3), for the coefficients (QK−m ) of vj−m in (2) , we get
QK−m =
m−1 X
"
K X
Θm−l−1 (E n
#
− Θn )Σn GK,l + [Q − Σ − Cj−m − R] GK,m
l=−2 n=1
+
K X
[Cj−m,j−l ] GK,l
l=m+1
(m = j − L, . . . , −2, −1, 0, . . . , K, . . . , j) .
(A.5)
Also, for m = −2, −1, 0, . . . , K, substituting Cj−m = C and Cj−m,j−l = Cj−l+l−m,j−l = C(E − Φ)Φl−m−1 + R(E − ∆)∆l−m−1 in (A.5), we get, 22
QK−m =
m−1 X
"
K X
Θm−l−1 (E n
#
− Θn )Σn GK,l + [Q − Σ − C − R] GK,m
l=−2 n=1 K h X
+
i
C(E − Φ)Φl−m−1 + R(E − ∆)∆l−m−1 GK,l
l=m+1
(m = −2, −1, 0, . . . , K)
(A.6)
Using the above, the required Ql ’s can be computed easily as described in the subsection below. Notice the above Ql ’s in equation (A.6) are j- independent. The other coefficients, i.e. those of vj−K−1 , vj−K−2, . . . , v0 and of vj+3 , vj+4 , . . ., can be shown to be zero. This is illustrated for K = 2 in our technical report [15]. A general proof for the same is given in the section B.
A.1
Computation
After obtaining FK,l ’s thus, GK,l (l = −2, −1, . . . , K) can be computed from (A.4). Then, using them directly in (A.6), the required Ql (l = 0, 1, . . . , K + 2) can be computed. An alternate way of computing the GK,l ’s is by the following properties and recursion which are obtained from (11, 11,A.4). G1,−2 = Φ∆ ; G1,−1 = −(Φ + ∆) − Φ∆Θ1 ; G1,0 = E + (Φ + ∆)Θ1 ; G1,1 = −Θ1 ; Gk,l = 0 (l ≤ −3) ; Gk,l = 0 (l ≥ k + 1) ; Gk,l = Gk−1,l − Θk Gk−1,l−1 (2 ≤ k ≤ K , −2 ≤ l ≤ k)
(A.7)
Another interesting property of Gk,l ’s is,
k X
Gk,l =
l=−2
k X
Gk−1,l − Θk
l=−2
=
k−1 X
=
Gk−1,l−1
l=−2
Gk−1,l − Θk
l=−2
k X
k−1 X
l=−2
k−1 X
Gk−1,l−1
l−1=−2
(since Gk−1,k = Gk−1,−3 = 0)
Gk−1,l (E − Θk )
Also, from (A.7), we have
P1
l=−2
(A.8)
G1,l = (E − Φ)(E − ∆)(E − Θ1 ). Hence, we 23
get the following result combining the above results: k X
Gk,l = (E − Φ)(E − ∆)
(E − Θn )
(A.9)
n=1
l=−2
B
k Y
Proof by Mathematical Induction
It is possible to prove the results in the previous section for any K, by Mathematical Induction. It is done as follows. From (A.4, A.7), we have,
Gk+1,l = Gk,l − Θk+1 Gk,l−1 ; Gk+1,l = 0 , if l ≤ −3 or l ≥ k + 1 .
(B.1)
Since, K itself is arbitrary in this section, let the Ql ’s be designated differently (k) to take that into account. Let Qk−m (m = j − L, j − L + 1, . . . , j) be the Ql ’s of < j >(2) when only the first k customer arrival streams are present and others are absent. Theorem 1 Referring to equation (A.5) for the row j (c+K +1 ≤ j ≤ L−3), (K) for all K, QK−m = 0 (j − L ≤ m ≤ −3). Proof Assume the proposition is true for any K=k. Hence, from (A.5), for the range j − L ≤ m ≤ −3, we have
(k)
Qk−m =
k X
Cj−m,j−l Gk,l
l=m+1
=
k X
Cj−m,j−l Gk,l (since Gk,l = 0 if l ≤ −3)
l=−2
= 0 (j − L ≤ m ≤ −3) .
(B.2)
For K = k + 1, from (A.5), we get 24
(k+1)
Qk+1−m =
k+1 X
Cj−m,j−l Gk+1,l
l=m+1
=
k+1 X
Cj−m,j−l Gk,l − Θk+1
l=m+1
k+1 X
Cj−m,j−l Gk,l−1
l=m+1
(substituting (B.1))
= 0 − Θk+1
k+1 X
Cj−m,j−l Gk,l−1 (since Gk,k+1 = 0 & using (B.2))
l=m+1
= −Θk+1
k X
Cj−(m−1)−1,j−(l−1)−1 Gk,(l−1) (rearranging)
l−1=m k X
= −Θk+1
Cj−(m−1),j−(l−1) Gk,(l−1)
l−1=(m−1)+1
(since Gk,m = 0 & Cj−(m−1)−1,j−(l−1)−1 = Cj−(m−1),j−(l−1) ) = 0 (comparing with (B.2)) . Hence the proposition is true for K = k + 1. Also, the proposition has been proved for K = 1 in [17] and for K = 2 in [15]. Hence, the theorem is true for all values of K ≥ 1. 2 Theorem 2 Referring to equation (A.5) for the row j (c+K +1 ≤ j ≤ L−3), (K) for all K, QK−m = 0 (K + 1 ≤ m ≤ j). Proof Assume the proposition is true for any K = k. Hence, from (A.5), we have
(k) Qk−m
=
m−1 X
"
k X
Θm−l−1 (E n
#
− Θn )Σn Gk,l (k + 1 ≤ m ≤ j)
l=−2 n=1
=
k X
"
k X
Θm−l−1 (E n
#
− Θn )Σn Gk,l = 0
l=−2 n=1
(B.3)
(since Gk,l = 0 for l > k) .
Then, for K = k + 1, writing down the expression for (k+1)
Qk+1−m from (A.5), substituting (B.1) as before and expanding the terms, we get,
(k+1) Qk+1−m
=
k+1 X
"
k X
Θm−l−1 (E n
− Θn )Σn +
Θm−l−1 (E k+1
− Θk+1 )Σk+1 Gk,l
l=−2 n=1
−Θk+1
k+1 X
"
k X
Θm−l−1 (E n
l=−2 n=1
25
− Θn )Σn +
#
Θm−l−1 (E k+1
(B.4) #
− Θk+1 )Σk+1 Gk,l−1
Using (B.3) and Gk,k+1 = 0 in the above, it simplifies to,
(k+1) Qk+1−m
=
k+1 X
Θm−l−1 (E − Θk+1 )Σk+1 Gk,l k+1
l=−2 k X
−Θk+1
"
k X
Θ(m−1)−(l−1)−1 (E n
− Θn )Σn Gk,l−1
l−1=−3 n=1 k X
−Θk+1
#
Θm−l−1 (E − Θk+1 )Σk+1 Gk,l−1 . k+1
l−1=−3
However, the middle term of the r.h.s. above would be 0 for k + 1 ≤ m − 1 ≤ j by comparing with (B.3) and by using Gk,−3 = 0. Hence, we get, for k + 1 ≤ m − 1 ≤ j and hence for k + 2 ≤ m ≤ j,
(k+1)
Qk+1−m =
k+1 X
Θm−l−1 (E − Θk+1 )Σk+1 Gk,l k+1
l=−2
−
k X
m−(l−1)−1
Θk+1
(E − Θk+1 )Σk+1 Gk,l−1
l−1=−2
= 0 (m = k + 2, k + 3, . . . , j) (since Gk,k+1 = 0) .
(B.5)
Hence, the proposition is true for K = k + 1. The proposition has already been proved for K = 1 in [17] and for K = 2 in [15]. Hence, the theorem is true. 2 Theorem 3 Referring to equation (A.5) for the row j (c+K +1 ≤ j ≤ L−3), (K) for all K, QK−m (m = −2, −1, 0, . . . , K) are j-independent. Proof in (A.6). From
(K)
QK−m for m = −2, −1, 0, . . . , K are separately derived
(K)
the r.h.s. of (A.6), it is clear that QK−m (m = −2, −1, . . . , K) are j- independent. 2 Theorem 4 Referring to equation (A.5) for the row j (c+K +1 ≤ j ≤ L−3), P (K) for all K, K m=−2 QK−m is singular. Proof Assume the theorem is true for some K = k. The expres(k) (k+1) sions for Qk−m and Qk+1−m are, 26
(k) Qk−m
=
m−1 X
"
k X
Θm−l−1 (E n
#
− Θn )Σn Gk,l + Q −
l=−2 n=1
+
k h X
"
k X
#
Σn − C − R Gk,m
n=1
i
C(E − Φ)Φl−m−1 + R(E − ∆)∆l−m−1 Gk,l
l=m+1
(m = −2, −1, 0, . . . , k)
(k+1) Qk+1−m
=
" m−1 X k+1 X
Θm−l−1 (E n
#
− Θn )Σn Gk+1,l + Q −
l=−2 n=1
+
k+1 X
l=m+1
h
"
k+1 X
#
Σn − C − R Gk+1,m
n=1
i
C(E − Φ)Φl−m−1 + R(E − ∆)∆l−m−1 Gk+1,l
(m = −2, −1, 0, . . . , k + 1)
Substituting Gk+1,l = Gk,l − Θk+1 Gk,l−1 in the latter, we get,
(k+1)
Qk+1−m =
m−1 X
l=−2
+
h
i
Θm−l−1 (E − Θk+1 )Σk+1 Gk+1,l k+1
m−1 X
"
k X
#
Θm−l−1 (E − Θn )Σn (Gk,l − Θk+1 Gk,l−1 ) n
l=−2 n=1
"
−Σk+1 Gk+1,m + Q − +
k+1 X
l=m+1
h
k X
#
Σn − C − R (Gk,m − Θk+1 Gk,m−1 )
n=1
i
C(E − Φ)Φl−m−1 + R(E − ∆)∆l−m−1 (Gk,l − Θk+1 Gk,l−1 )
After expanding, rearranging and regrouping the terms, we get
(k+1)
(k)
(k)
Qk+1−m = Qk−m − Qk−(m−1) Θk+1 +
m−1 X
h
i
(E − Θk+1 )Σk+1 Θm−l−1 Gk+1,l k+1
l=−2
−Σk+1 Gk+1,m (m = −2, −1, . . . , k + 1) (k)
(k)
= Qk−m − Qk−(m−1) Θk+1 +(E − Θk+1 )Σk+1
m−1 X
l=−2
h
Θm−l−1 Gk+1,l k+1
i
−Σk+1 Gk+1,m (m = −2, −1, . . . , k + 1) Then, summing up the terms from m = −2, . . . , k + 1, we get 27
k+1 X
(k+1) Qk+1−m
m=−2
=
k X
m=−2
(k) Qk−m [E
+Σk+1
m−1 X
− Θk+1 ]
Θm−l−1 Gk+1,l k+1
−
l=−2
m X
l=−2
Θm−l k+1 Gk+1,l
.
By substituting Gk+1,l = Gk,l − Θk+1 Gk,l−1 in the r.h.s. of the above equation and expanding, the terms other than the first term cancel off, leaving, k+1 X
m=−2
(k+1) Qk+1−m
=
k X
m=−2
(k) Qk−m [E
− Θk+1 ] .
(B.6)
The above r.h.s. expression is clearly a singular matrix if the theorem is true hP i (k) k for K = k, that is, m=−2 Qk−m is singular, since [E − Θk+1 ] is a diagonal matrix. The theorem is proved for K = 1 in [17] and for K = 2 in [15]. Hence the theorem is true for any K.
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