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Performance analysis of an integrated hybrid-switched multiplex structure
81
Performance Analysis of an Integrated Hybrid-Switched Multiplex Structure * R . H . Kwong and A. Leon-Garcia Department of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received 10 August 1981 Revised July 1982, 9 February 1984
The performance of an integrated voice/data hybrid-switched multiplex structure is analyzed. The approach is based on an imbedded two-dimensional Markov chain associated with the voice and data queueing processes, which accounts for their interaction. Using generating functions, a method for determining exactly the average data delay is given. As an application, an analytical expression for the average data delay is derived for the so-called single channel case. These results should be considered primarily as a theoretical contribution since the numerical difficulties involved in the solution procedure for the general case are formidable.
Keywords: Integrated Voice/Data Networks, Movable Boundary Hybrid Switching, Data Delay Performance, Two-dimensional Imbedded Markov Chains. Raymond H. Kwong was born in Hongkong on September 20, 1949. He received the S.B., S.M. and Ph.D. degrees in Electrical Engineering from the Massachusetts Institute of Technology, Cambridge, in 1971, 1972 and 1975, respectively. From 1975 to 1977 he was a Visiting Assistant Professor of Electrical Engineering at McGill Universty and a Research Associate at the Centre de Recherches Mathematiques, Universit6 de Montr6al, Montreal, Canada. Since August 1977 he has been with the Department of Electrical Engineering at the University of Toronto, where he is now an Associate Professor. His current research interests are in the areas of estimation and stochastic control, identification and adaptive control, and computer communication networks. Dr. Kwong is a member of Sigma Xi, Tau Beta Pi and Eta Kappa Nu.
Alberto Leon-Garcia was born in Mexicali, Mexico on April 5, 1952. He received the B.S., M.S. and Ph.D. degrees in Electrical Engineering from the University of Southern California in 1873, 1974 and 1976, respectively. During the 1976-77 academic year he was a Visiting Assistant Professor in the Department of Electrical Engineering at the University of Maryland, College Park. Since August 1977 he has been with the Department of Electrical Engineering at the University of Toronto, presently as an Associate Professor. His current research interests are in universal source coding, integrated voice-data networks and line coding. He is currently Editor of the IEEE Information Theory Group Newsletter and an Associate Editor of the IEEE Transactions on Communications.
1. Introduction
Current developments in communication technology indicate that the variety and volume of traffic will i n c r e a s e s h a r p l y in the n e a r future. To m e e t these d e m a n d s , future n e t w o r k s m u s t be able to p r o v i d e * This work was supported in part by the Department of Communications, Canada, under Contract OSU79-00041, and by the Natural Sciences and Engineering Research Council of Canada under Grant No. A0875. North-Holland Performance Evaluation 4 (1984) 81-91 0166-5316/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
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R.H. Kwong, A. Leon-Garcia / Integrated hybrid-switched multiplex structure
efficient use of trunk capacity for a wide variety of transmission characteristics such as voice and interactive data. The integration of voice and data transmission in a common rather than two separate systems allows switching equipment as well as transmission capacity to be shared, thereby reducing expenses for the entire network [1]. Another advantage to integrating voice and data is the capability of providing interconnection between the broadest possible community of subscriber terminals. These factors have led to a growing interest in integrated networks. Various schemes of integration have been proposed and studied in a number of papers dealing with the modelling and analysis of such networks [1-7]. In this paper we focus our attention on the performance analysis of the so-called movable boundary hybrid switching scheme discussed in [4]. In. this scheme, both circuit and packet switching are provided by the network through a special time division multiplexing format whereby a frame of constant duration is divided into two compartments, one dedicated to circuit-switched traffic and the other to packet-switched traffic. The frame duration is the same throughout the network in order to provide a nearly synchronous virtual path to circuit-switched traffic. The compartments themselves are subdivided into slots of equal size. The voice digitization rate and the data packet size are assumed to be such that each voice or data customer will seize one voice or data slot per frame. The motivation for this switching scheme is to match the switching method to the traffic type. Thus voice traffic is circuit-switched while data is packet-switched, thereby achieving integration of the two types of traffic. The term movable boundary refers to the following method of designing the compartments. Data traffic is allowed to use idle time slots in the circuit-switched compartment. The sizes of the circuit-switched and packet-switched compartments may thus vary from frame to frame, hence the name movably boundary. Although this increases the complexity of the multiplexer, it is hoped that the utilization of the channel will be enhanced. The performance of the voice traffic is analyzed using the probability of blocking criterion, while that of data traffic is analyzed using the average delay criterion. More detailed discussions of the movable boundary hybrid-switched integration scheme are given in [2,4]. The first analysis of the movable boundary hybrid-switched scheme was given by Kummerle [3]. He did not attempt to characterize the performance exactly, but gave an approximate method whose accuracy was checked by a simulation model. The first attempt at an exact analysis was given by Fischer and Harris [4], who took the frame structure explicitly into account. Another approximate analysis was also given by Occhiogrosso et al. [5]. These results were used to make design calculations for feasibility studies in integrated networks [11]. It turns out that the 'exact' analysis described in [4] is incorrect in so far as the calculation of the average data delay is concerned. This was suggested implicitly in a thesis by Chang [8], and later explicitly by Weinstein et al. [9]. Chang [8] gave an exact analysis based on two-dimensional birth-death processes, effectively ignoring the frame structure. Weinstein et al. [9] pointed out explicitly where the error in [4] lies and gave simulation results to illustrate the discrepancy between the predicted data delay from the results of [4] and the data delay obtained from simulations of the model. Only in the very special, so-called single channel case does the method of Chang give rise to an analytical solution [9]. In general, numerical methods must be used and the numerical difficulties have also been pointed out in [8]. In this paper we present an exact analysis of the integrated switch performance based on the model given in [4] that takes into account the frame structure. Our method uses properties of the two-dimensional Markov chain associated with the voice and data queueing processes and accounts for their interaction. The paper is organized as follows. In Section 2 we briefly describe the Fischer and Harris model for the integrated switch. In Section 3 we show that the generating functions associated with the data process satisfy a linear matrix equation with the right hand side containing a certain number of unknowns. In Section 4 we show how these unknowns may be determined from finding the roots of an exponential polynomial. These results are then specialized to the single channel case in Section 5, and an analytical formula for the average data delay is derived. As in the exact analysis given in [8], these results are primarily of theoretical interest since the numerical difficulties associated with the solution (except for the single channel case) are formidable. A computationally simple approximate analysis which gives accurate estimates of the average data delay has been obtained and is reported elsewhere [10,14]. Some of these results have appeared previously in report form [10].
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2. Brief description of the queueing model In this section we give a concise description of the queueing model formulated in [4] which we shall use for performance analysis. The movable boundary integration scheme can be modelled as a queueing system with two types of arrivals, voice and data, and an operating rule that allows these customers access to the system at fixed intervals of time (referred to as opening the gate). An arriving voice customer waits in a buffer until the next opening of the gate. If the number of free voice slots is greater than the number of voice customers ahead of him, he receives service. If not, he is lost and leaves the system. For the data traffic, arrivals are buffered, and, at the opening of the gate, are placed on the data slots and any unoccupied voice slots on a first come, first served basis. We assume both the voice and data buffers have infinite capacity, although the finite data buffer case can also be readily handled. The queueing problem to be addressed may now be formulated as follows: given the traffic characteristics, find the blocking probability for voice traffic, and the average delay for data traffic. We adopt the following assumptions for the voice and data processes (the notation is that of [4]). Voice and data customers arrive in two independent Poisson processes with parameters )~ and 0, respectively. The holding time distribution for voice is a negative exponential distribution with mean 1 / # . Data packets are assumed to be of constant size, and a data packet is processed within a frame period, assumed to be of duration b, so that the service distribution function Fd(x), is of the form Fd(X ) = 0 for x < b and Fd(x ) = 1 for x >i b. The numb'er of voice and data slots in a frame is assumed to be N v and No, respectively, with N = N v + N o. We assume that all parameters are expressed in" consistent units.
3. The two-dimensional imbedded Markov chain and conditional generating functions The analysis of the voice traffic performance in terms of the blocking probability has been successfully accomplished in [4], the results of which we quote for completeness later in this section. In order to analyze the data traffic performance, we need to use a two-dimensional imbedded Markov chain to characterize the joint voice and data queueing process. To do so, let us definep v to be the probability of i voice arrivals in a duration of length b, qVj the conditional probability o f j busy voice slots just before the opening of the gate given that there were k present just after the last opening, and pD the probability of i data customer arrivals in a duration of length b. By our assumptions, we have
pV = e xh()~b)'/i!,
i = 0 , 1, 2 . . . . .
q ~ = ( ( ~ )(1-e-~'b)k-j(e-t'b)j' [ 0,
(3.1)
Nv>~k>~J'
(3.2)
otherwise,
pY = e-°b(Ob)'/i!,
i = 0, 1, 2 ....
(3.3)
Let L v be the steady state number of voice customers occupying voice slots just before the gate opens. It is shown in [4] that the voice queueing process is stable when Ob < N - E ( L v)- Furthermore, the steady state distribution of L v, ~rv, i = 0, 1, 2 . . . . . N v is determined by the equations Nv
C = E ¢rVpv,
J = 0,1 . . . . . Nv,
(3.4)
iffi0
and the normalization condition Nv
7rv = 1, j=0
(3.5)
R.H. Kwon& A. Leon - Garcia / Integrated hybrid- switched multiplex structure
84
where pV are the transition probabilities given by Nv--1
z
vv v qkjPk-, + qNvj
k~max(i,j} __
V
--qNvj
pV
for
k~Nv-i
i = 0 , 1 .... ,N v - 1, j = 0,1 . . . . . Nv
f o r i = N v , j = 0 , 1 . . . . . N v.
(3.6)
Finally, the blocking probability PL for voice is given by
kpV+(Nv-j)1-
P L = I - - ~ - - ~ , j=0 7rv[ ~ 1
k=0 ~ pV
+~rVv_l(l_pV)
.
(3.7)
Although, in general, the evaluation of PL requires the solution of the matrix equation (3.4) subject to the normalization condition (3.5), it is shown in [4] that the blocking probability PL is well approximated by the Erlang B formula [11] provided Xb is sufficiently small, which indeed is the case in applications. We now turn to the analysis of data traffic. Let QV and QD be the number of voice and data customers, respectively, in the channel immediately after the t th opening of the gate, with the corresponding steady state variables denoted by Qv and QD- We shall derive the transition probabilities defined by the two-dimensional Markov chain given by the joint process (QV, QD). First, it is easy to write down the following conditional probabilities: min{i,m }
pr{QV=mlQV_l=i,
QtO_l=j } =
~_,
v v qikP,,-k,
m = 0,1 . . . . . N v - 1,
(3.8)
k=0
Pr{QV = NvlQV_l = i, Q~_, = j } =
~o[qVr~v
(3.9)
pv-k ]
and Pr{QD = n I Q v = m,QV_a = i,QD_l = j } =
p~
ifj~< N - i - 1,
P~+N - i - j
i f O <~i + j - N <~n .
(3.10)
From the above equations (3.8)-(3.10) we see that the joint process (QV, Q D) defines a homogeneous two-dimensional Markov chain with transition probabilities Pr(Q v = m, Q,D = n Mo V l = i, 0,°_1 = j ) & P i j , m n given by
=
Pij,mn
~--"
m i n { i, m }
~ v v D qik Pro- k P, k=0 min{i,m} E
k=,
= E k~0
V
V
r
--~N pv-k pD v
qik ~
=
k=O
D
qikPm-kPn+N-i-j
r=N v
Pr-k
J
P2+N-i-j
for
for
j<~N-i-1, m < Nv, 0 ~< i + j -- N ~
for j < ~ N - i - 1 , m =Nv, for
m = N v.
(3.11)
Let the state probabilities of the Markov chain, Pr(Q v = m, Q o = n), be denoted by P ' , . P'n then satisfies the equation t __ P,~, - ~_,pij,,,,P,jt - 1 ,
i,j
(3.12)
R.H. Kwong, A. Leon-Garcia / Integrated hybrid-switched multiplex structure
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where the summation is over i,j satisfying i,j >1O, i + j <~N+ n. Let P ~ ( z ) = E,~=oPt,z ". We shall call P,,t,(z), m = 0, 1 . . . . . Nv, the conditional generating functions of the data process. F r o m (3.12) we see that
P~,(z) = Y'~ ~ t
P'-lz". ij
Pij,mn
(3.13)
i , j n=O
The right-hand side of (3.13) may be simplified by using (3.11). We find min(i,m} E
Pij,mn zn
=
j ~< N - i - 1,
E
n=0
k=0 min{i,m } =
E
e-Ob(l-z)
V P,, V- ~ e-0b(l qik
k=0
=
V Pm v- k qik
,~v ~
for
-Z)zi+j-N
for
,,v
/'~ik Z.., t,r-k/e-°b~l-z) r=U v
k=O [
=
for
j
qV E PVk e-Ob('-:)Zi+J-N r= Nv J
k=0 [
m < Nv, 0 ~< i + j - N, m < Nv,
j<~N-i-1 m = Nv,
'
for
(3.14)
m = N v.
Substituting (3.14) into (3.13) we obtain, after some computation, that Nv N - i - 1
P~(z) = E i=0
min{i,m}
E
E
j=0
k=O
q,,Vp,.V_ e - e e ( , - z , p , ; - , ( 1 _ z , + j - N )
Nv min( i,m }
+ Y~,
E
i~0
v v e-Ob~l-z)p/- l (z) zi-N q~kPm-k
f o r m = 0 , 1 ..... Nv
1,
(3.15)
k=0
Nv N - i - 1
Puv(Z) = E
i
E
i=0
E qV ~, pVkpij-i e-Ob(1-Z,[l _zi+j-N]
j=0 Nv
r=N v
k=0
i
+ 2
2 qV ~ pV_kpit-l(z)e-Ob¢,-Z)zi-N"
(3.16)
r=N v
i=0 k=0
Using (3.15) and (3.16) it is now easy to characterize the steady state conditional generating functions that will be used later in the evaluation of the average data delay. Let %j denote the steady state probability of having i voice customers a n d j data customers in the system, and let the steady state conditional generating function be denoted by ~r,,(z)= Z,~__0~r,,,,,z", m = 0, 1 . . . . . N v. Assuming that the stability condition for the queueing process holds, we have lim,~ ~ P,,', t -_ ~rm,. On taking limits in (3.15) and (3.16) we finally obtain the following equations for the conditional generating functions ~rm(z): Nv N - i - 1
~r,,,(z)= E i~O
min{i,m}
E
E
j=O
k =0
qikP,,-kv v
e-Ob(a-z)%j(l_zi+y-N )
N v rain{i,m)
+E
E
i~0
vv q~kPm-k e-Ob"-%r~(z)z '-N,
~'Nv(Z) = E
q,,v E
E
i=0
m=0,1,.
, N v - 1,
(3.17)
k=O
j=O
k=O 1.
r=N v
v Pr-k ~ri/ e-0t,~l-z)(1 _ Z,+j-N)
j
Vo[qVvpV_]
+ Y'~ i=0
%(z) e-Ob('-~)zi-U.
=
=
(3.18)
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R.H. Kwong, A. Leon - Garcia / Integrated hybrid- switched multiplex structure
It will be convenient to re-write (3.17) and (3.18) in matrix form. Define the vector of generating functions ~r(z) T = [%(z), ~h(z) . . . . . ~ruv(z)], where x T denotes the transposed of the vector x. It is clear, by taking terms involving ~r~(z) on the right-hand side to the left, that (3.17) and (3.18) may be written as a vector-matrix equation in 7r(z) as
A(z)~r(z) = b(z),
(3.19)
where A(z) is a known ( N v + 1) × ( N v + 1) matrix consisting of coefficients multiplying the ~r,(z)'s in (3.17) and (3.18), while b(z) is a vector containing the unknowns ~rij, i= 0 ..... N v , j = O, 1..... N - i - 1. To solve (3.19) for rr(z), we must obtain a total of N + ( N - 1)+ . . . + ( N - N v ) = ½(N v + 1)(2N - Nv) N auxiliary equations to determine the unknown ~%'s in b(z). Once ~r(z) is determined, it is then straightforward to calculate the average data delay E(Wo). Thse matters will be dealt with in the next section.
4. Determination of the average data delay In this section we shall derive auxiliary equations for determining the generating function 7r(z), and then use it to compute the average data delay. The method we shall use is based on the following observation. Let Ai(z ) be the matrix obtained from A(z) by replacing the ith column by the vector b(z). Then Cramer's rule says that ~r,(z) --- det Ai(z)/det A(z). But ~r~(z), being a generating function, must be analytic in Izl ~< 1. Hence any zeros of det A(z) in Izl ~ 1 must be cancelled by the corresponding zeros in det At(z ). Such an observation is often invoked in queueing theory [11] and our main task here is to show that this generates the desired number of equations to determine the unknowns in b(z). To do this we first re-write det A(z) in a convenient form for analysis. Let a,g(Z) denote t h e / j t h element of the matrix A(z). Define rain( j,m }
a,,j(z)
=
~-,
qjkVPmV-ke-Ob~l-Z)zj - u
f o r m = 0 , 1 . . . . N v - 1,
k=0 J
-- Y'~ qVk ~ pV k e-Ob'l-Z)Z j - u k~O
for m = N v.
(4.1)
r=N v
We then have the relations
aij(z)---aij(z)
for i ~ j ,
aii(z)=l-aii(z).
(4.2)
From the definition of a~j(z) it is easy to verify that Nv (4.3)
E Otij(Z) = e-Ob(l-z)zJ-N i~O
(cf. (3.8) and (3.9)). Furthermore, in the j t h column of A(z) every element contains a factor of the form z j-N except t h e j j t h entry. From this observation, we may write det A ( z ) = z -tE~'-~°(N-j'ld e t . 4 ( z ) = z - ~ det ,,l(z),
(4.4)
where A(z) is a matrix having as entries
~ij(z)=-~tij(z)
for i . j ,
~.(z)=zU-l--rii(z )
(4.5)
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87
with min ( i, j }
aij(z)
=
E
qVkpV-ke-ab(1-~)
for i = 0, 1 . . . . . N v - 1,
k=O
J
= y'qVk ~ k=O
pVke-Ob"-~)
for i = N v.
(4.6)
r=N v
Thus the zeros of det A(z) in Izl ~ 1 correspond to those of.~(z) in Izl ~ 1. The number of zeros of A-(z) in Izl < 1 is established by the following proposition.
Proposition 4.1. The number of zeros
o f d e t ,,l(z) in [z~ <~1 is N.
Proof. It is easy to see from (3.8) and (3.9) that Nv
(4.7)
E ~¢ij(z) = e-Ob(l-z)" i=O
Introduce the matrix A(z) whose elements are given by
(4.8) so that
?q,( Z ) = zN -- z' ~,( Z ), 3ij(Z)=--ZJaij(Z),
(4.9)
i4=j.
(4.10)
We may therefore write (4.11) where
- oo(Z)
Z o (Z)
...
zN
O v(Z)
...
zN
l v(z)
B(z) =
(4.12)
aNvo(Z)
ZaNv,(Z)
---
zNva v v(Z)
Note that det .~(z) =
z~J~'Jdet i f ( z ) ,
(4.13)
so that the total number of zeros of det A(z) in Izl ~ 1 is equal to those of det A-(z) plus E~Y~1j zeros at the origin. Now, for Izl ~ 1, an application of Gershgorin's theorem [12] shows that the eigenvalues of B(z), )%(z), i = 0 . . . . . N v, have magnitude I~,,(z)l ~ 1. Thus, from (4.11), we find that if the eigenvalues Xi(z ) are all distinct, the roots of det A(z) are given by solutions of the equations
zU=)~i(z),
i = 0 , 1 . . . . . N v.
(4.14)
An application of Rouch6's theorem to (4.14) shows that for each i = 0, 1 . . . . . N v there are N solutions of (4.14) in Izl < 1. Hence there are a total of ( N v + 1)N zeros of det A(z) in Izl < 1. Thus the total number of
R.H. Kwong, A. Leon- Garcia / Integrated hybrid-switched multiplex structure
88
zeros of A ( z ) in Izl ~< 1 is ( N v + 1)N-Y~, j=lJ uv "= .~, according to (4.13). If the eigenvalues are not all distinct, we can apply the same perturbation argument as in [13] to show that we have the desired number of zeros. This proves the proposition. The following corollary is immediate from (4.4). D
Corollary 4.2. The number of zeros o f d e t A( z ) in tzl <~1 is N. Observe that one of the zeros of det A ( z ) is z = 1. If we denote the other zeros by sp, O = 1 ...... N - 1, we then have the equations d e t A i ( s p ) = 0,
p = 1 . . . . . , N - 1.
(4.15)
These together with the normalization condition ~Irij = 1
(4.16)
i,j
yield the desired number of equations which determine the unknown %j's in b(z). Thus, we may now solve for the conditional generating function tr(z) using (3.19). Once ~r(z) has been determined, the average number of data customers E ( Q D ) is found by Nv
E(QD) = ~ i=0
..1
(4.17)
~zCri(z) z= 1
The average data delay E(WD) is obtained by application of Little's formula [11] to give E ( W D ) = ½b + o E ( Q D ) ,
(4.18)
where the first term in the right-hand side of (4.17) represents the delay due to the gate in the frame structure, and the second term represents the delay once access has been gained to the system. While the above procedure in principle enables us to determine the average data delay, an accurate computation will be a very difficult numerical problem. First of all, A ( z ) is a full matrix with entries being transcendental functions of z, with no special properties that can be exploited. Computing its determinant accurately for moderate values of N v, say around 20, is already known to be a difficult task. After we have evaluated det A(z), which takes the form of an exponential polynomial, we still have to determine its N zeroes in Izl ~< 1. The numerical difficulties involved, even in the case where the exponential factors are absent, have been pointed out by Chang [8]. Our situation would of course be worse in general. Thus the above analysis should be treated primarily as a theoretical contribution to the understanding of the integrated v o i c e / d a t a link model. There is, however, an analytical expression that can be derived for the average data delay in the simplest possible case N v = 1, N D = 0, referred to as the single channel case. This will be presented in the next section. The derivation of this result will also serve to illustrate the procedure discussed above for the computation of the average data delay.
5. The average data delay in the single channel case In this section we specialize the results of Section 4 to the single channel case where N v = 1, N D = 0. This case has also been considered by Weinstein et al. [9] with, however, the crucial difference that they have to assume exponential distributed data packet lengths. The resulting equations in their case are substantially simpler, and involve no exponential terms. The somewhat more intricate calculations required for the determination of average data delay in the constant data packet case will now be presented.
R.H. Kwon& A. Leon-Garcia / Integrated hybrid-switched multiplex structure
89
We first write down the equation satisfied by the conditional generating function. Eq. (3.19) for 7r(z) in this case becomes
- ( l - p V ) e - O b . - Z , z -' [ --
l_(l_q'~opV)e-Ob~-z,
pV e-Ob'l-z)~roo(1--z-1) (1 _ p ,) 0o e _ O b , _ Z , ( l _
[~rl(z )
]
z_,)
•
(5.1/
There is only 1 root inside Izl ~< 1, and it is z = 1. We need only the normalization condition Y'.~r, (1)=Evrig = 1 i
(5.2)
i,j
to determine the unknown quantity %o in (5.1). Solving for %(z) and %(z) we obtain, after some computations, % ( z ) = p v e - o b ' l - ~ ' % o ( z - 1 ) [ 1 - q v l e oh,,-~)]
~v _v~~ 20h(1-~) } × ( z [ 1 - e - ° b ( 1 - z ) ] - p v e - O h ( l - z ) + zqVop v e -°ml-z) +'tllt'o
1
,
(5.3)
¢rl( g ) ='n'oo e-°b(l-z'( g - 1)(1 _pV)
× (z[1-e-°ba-z']--pVe-°b"-z)+zqVopVe-°b'l-z'
+ qupoV v e - 2 0 b . - = ) ) - I
(5.4)
Since the numerators and denominators of %(z) and ~h(z) all vanish at z = 1, we need to apply L'Hospital's rule to evaluate %(1) and %(1). We find
v v v v + qa, v Pov Ob - Ob), % (1) = Poqm%o/(qmPo V
V
V
% (1) = %o (1 - Po )/(qloPo + qVl P vOb - Ob).
(5.5) (5.6)
Using the normalization condition (5.2) we get
%o = ( qlV p~ + qVlpVOb - Ob)/(1 - pV + qVopV ).
(5.7)
Eqs. (5.3), (5.4) and (5.7) completely determine % (z) and % (z). Now the generating function for the steady state data customers QD is given by
a~(z) =
E~,(z) i
so that in this case
Go( z ) = %o e-°b'l-~)( z - 1)[1 -PoqnV v e-O/,(l-~)] X{z[l_e_Obo_z)]_pV
e_Obd_~)+zqVopV e_Ob(l_~,+qVlpV e_2Obo_~) } 1.
(5.8)
The average number of data customers is then given by d E(QD) = ~zz GD(Z)~=1"
(5.9)
Let u(z) and v(z) be the numerator and denominator of GD(Z), respectively. G'D(z) will be indeterminate at z = 1 since its numerator as well as denominator vanish at z = 1. Applying L'Hospital's rule to G~( z ) gives G;(1) = 0'(1) u"(1) - u'(1) v"(1)
2[v,(1)] 2
(5.10)
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R.H. Kwon& A. Leon-Garcia / Integrated hybrid-switched multiplex structure
2800
2400
2000
o 0) m
1600
m
1200
800
400
10
20
30
40
SO G (packets/sec)
Fig. 1. Average data delay as function of data load (constant data packets): N v = 1, N D = 0, ~ = 0.01, # = 0.01, b = 0.01.
Computation then yields EQD = [2(qV0pV + qVlpVOb - Ob) %o8b(1 - 2qVlp v) - %o (1 - pVqV1) Ob(2qVo pV + 3qVl pVOb _ Ob - 2)][ 2( qV1pVOb + qlv pV _ 85 )2] -1 (5.11) Finally, the average data delay is given by (4.18). As an example, for X = 0.01, /~ = 0.01, b = 0.01, the average data delay is plotted against 0 in Fig. 1. Notice that from the approximation using the Erlang B formula, E ( L v ) = 0 . 5 so that the stability condition for the data queue is 0b < 0.5. This is shown clearly in Fig. 1. When compared with the results of Weinstein et al. [9] for the single channel case using the assumption of exponentially distributed packet lengths rather then constant packets lengths considered here, we see that the results are very close, as would be expected. 6. Conclusions
We have presented an exact analysis of the integrated hybrid-switched voice/data multiplex structure previously considered by Fischer and Harris, using the method of two-dimensional imbedded Markov chains and conditional generating functions• The determination of the average data delay requires finding
R.H. Kwong, A. Leon -Garcia / Integrated hybrid-switched multiplex structure
91
the roots of an analytic f u n c t i o n of a complex variable within the u n i t disc, a n d the solution of a v e c t o r - m a t r i x equation. While the p e r f o r m a n c e of the integrated switch m a y thus in principle be evaluated, it is evident from the solution procedure that the required c o m p u t a t i o n s will be too complex to carry out in practice except in extremely simple cases. The results are therefore primarily of a theoretical n a t u r e a n d a suitable a p p r o x i m a t e p e r f o r m a n c e analysis is needed to u n d e r s t a n d the traffic b e h a v i o u r in realistic situations. O n e such a p p r o x i m a t i o n based o n a fluid a p p r o x i m a t i o n technique, which is c o m p u t a t i o n a l l y attractive a n d whose accuracy has been verified b y simulations, has been reported in [10,14].
References [1] I. Gitman and H. Frank, Economic analysis of integrated voice and data networks: A case study, Proc. IEEE 66 (1979) 1549-1570. [2] G. Coviello and P. Vena, Integration of circuit/packet switching by a SENET (Slotted Envelope Network) concept, Proc. National Telecommunications Conf., New Orleans, LA (1975) 42.12-42.17. [3] K. Kummerle, Multiplexer performance for integrated line and packet-switched traffic, Proc. 2nd Internat. Conf. on Computer Communications, Stockholm, Sweden (1974) 517-523. [4] M.J. Fischer and T.C. Harris, A model for evaluating the performance of an integrated circuit- and packet-switched multiplex structure, IEEE Trans. Commun. COM-24 (1976) 195-202. [5] B. Occhiogrosso, I. Gitman, W. Hsieh and H. Frank, Performance analysis of integrated switching communications systems, Proc. National Telecommunication Conf., Los Angeles, CA (1977) 12.4.1-12.4.13. [6] E. Arthurs and B.W. Stuck, A theoretical traffic performance analysis of an integrated voice-data virtual circuit packet switch, IEEE Trans. Commun. COM-27 (1979) 1104-1111. [7] E.A. Harrington, Voice/data integration using circuit
switched networks, IEEE Trans. Commun. COM-28 (1980) 781-793. [8] L.H. Chang, Analysis of integrated voice and data communication network, Ph.D. Thesis, Dept. of Electrical Engineering, Carnegie-Mellon Univ., 1977. [9] C.J. Weinstein, M.L. Malpass and M.J. Fischer, Data traffic performance of an integrated circuit- and packetswitched multiplex structure, IEEE Trans. Commun. COM-28 (1980) 873-878. [10] R.H. Kwong, A. Leon-Garcia and A.N. Venetsanopoulos, A study of a joint stochastic process for an integrated network, Final report, Prepared for the Department of Communications, Canada, under Contract OSU79-00041, 1980. [11] L. Kleinrock, Queueing Systems, Vol. I (Wiley, New York, 1975). [12] G. Strang, Linear Algebra and Its Applications (Academic Press, New York, 1976). [13] M.F. Neuts, A queue subject to extraneous phase changes, Adv. Appl. Probab. 3 (1971) 78-119. [14] A. Leon-Garcia, R.H. Kwong and G. Williams, Performance evaluation methods for an integrated voice/data link, IEEE Trans. Commun. COM-30 (1982) 1848-1858.
Performance Analysis of an Integrated Hybrid-Switched Multiplex Structure * R . H . Kwong and A. Leon-Garcia Department of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received 10 August 1981 Revised July 1982, 9 February 1984
The performance of an integrated voice/data hybrid-switched multiplex structure is analyzed. The approach is based on an imbedded two-dimensional Markov chain associated with the voice and data queueing processes, which accounts for their interaction. Using generating functions, a method for determining exactly the average data delay is given. As an application, an analytical expression for the average data delay is derived for the so-called single channel case. These results should be considered primarily as a theoretical contribution since the numerical difficulties involved in the solution procedure for the general case are formidable.
Keywords: Integrated Voice/Data Networks, Movable Boundary Hybrid Switching, Data Delay Performance, Two-dimensional Imbedded Markov Chains. Raymond H. Kwong was born in Hongkong on September 20, 1949. He received the S.B., S.M. and Ph.D. degrees in Electrical Engineering from the Massachusetts Institute of Technology, Cambridge, in 1971, 1972 and 1975, respectively. From 1975 to 1977 he was a Visiting Assistant Professor of Electrical Engineering at McGill Universty and a Research Associate at the Centre de Recherches Mathematiques, Universit6 de Montr6al, Montreal, Canada. Since August 1977 he has been with the Department of Electrical Engineering at the University of Toronto, where he is now an Associate Professor. His current research interests are in the areas of estimation and stochastic control, identification and adaptive control, and computer communication networks. Dr. Kwong is a member of Sigma Xi, Tau Beta Pi and Eta Kappa Nu.
Alberto Leon-Garcia was born in Mexicali, Mexico on April 5, 1952. He received the B.S., M.S. and Ph.D. degrees in Electrical Engineering from the University of Southern California in 1873, 1974 and 1976, respectively. During the 1976-77 academic year he was a Visiting Assistant Professor in the Department of Electrical Engineering at the University of Maryland, College Park. Since August 1977 he has been with the Department of Electrical Engineering at the University of Toronto, presently as an Associate Professor. His current research interests are in universal source coding, integrated voice-data networks and line coding. He is currently Editor of the IEEE Information Theory Group Newsletter and an Associate Editor of the IEEE Transactions on Communications.
1. Introduction
Current developments in communication technology indicate that the variety and volume of traffic will i n c r e a s e s h a r p l y in the n e a r future. To m e e t these d e m a n d s , future n e t w o r k s m u s t be able to p r o v i d e * This work was supported in part by the Department of Communications, Canada, under Contract OSU79-00041, and by the Natural Sciences and Engineering Research Council of Canada under Grant No. A0875. North-Holland Performance Evaluation 4 (1984) 81-91 0166-5316/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
82
R.H. Kwong, A. Leon-Garcia / Integrated hybrid-switched multiplex structure
efficient use of trunk capacity for a wide variety of transmission characteristics such as voice and interactive data. The integration of voice and data transmission in a common rather than two separate systems allows switching equipment as well as transmission capacity to be shared, thereby reducing expenses for the entire network [1]. Another advantage to integrating voice and data is the capability of providing interconnection between the broadest possible community of subscriber terminals. These factors have led to a growing interest in integrated networks. Various schemes of integration have been proposed and studied in a number of papers dealing with the modelling and analysis of such networks [1-7]. In this paper we focus our attention on the performance analysis of the so-called movable boundary hybrid switching scheme discussed in [4]. In. this scheme, both circuit and packet switching are provided by the network through a special time division multiplexing format whereby a frame of constant duration is divided into two compartments, one dedicated to circuit-switched traffic and the other to packet-switched traffic. The frame duration is the same throughout the network in order to provide a nearly synchronous virtual path to circuit-switched traffic. The compartments themselves are subdivided into slots of equal size. The voice digitization rate and the data packet size are assumed to be such that each voice or data customer will seize one voice or data slot per frame. The motivation for this switching scheme is to match the switching method to the traffic type. Thus voice traffic is circuit-switched while data is packet-switched, thereby achieving integration of the two types of traffic. The term movable boundary refers to the following method of designing the compartments. Data traffic is allowed to use idle time slots in the circuit-switched compartment. The sizes of the circuit-switched and packet-switched compartments may thus vary from frame to frame, hence the name movably boundary. Although this increases the complexity of the multiplexer, it is hoped that the utilization of the channel will be enhanced. The performance of the voice traffic is analyzed using the probability of blocking criterion, while that of data traffic is analyzed using the average delay criterion. More detailed discussions of the movable boundary hybrid-switched integration scheme are given in [2,4]. The first analysis of the movable boundary hybrid-switched scheme was given by Kummerle [3]. He did not attempt to characterize the performance exactly, but gave an approximate method whose accuracy was checked by a simulation model. The first attempt at an exact analysis was given by Fischer and Harris [4], who took the frame structure explicitly into account. Another approximate analysis was also given by Occhiogrosso et al. [5]. These results were used to make design calculations for feasibility studies in integrated networks [11]. It turns out that the 'exact' analysis described in [4] is incorrect in so far as the calculation of the average data delay is concerned. This was suggested implicitly in a thesis by Chang [8], and later explicitly by Weinstein et al. [9]. Chang [8] gave an exact analysis based on two-dimensional birth-death processes, effectively ignoring the frame structure. Weinstein et al. [9] pointed out explicitly where the error in [4] lies and gave simulation results to illustrate the discrepancy between the predicted data delay from the results of [4] and the data delay obtained from simulations of the model. Only in the very special, so-called single channel case does the method of Chang give rise to an analytical solution [9]. In general, numerical methods must be used and the numerical difficulties have also been pointed out in [8]. In this paper we present an exact analysis of the integrated switch performance based on the model given in [4] that takes into account the frame structure. Our method uses properties of the two-dimensional Markov chain associated with the voice and data queueing processes and accounts for their interaction. The paper is organized as follows. In Section 2 we briefly describe the Fischer and Harris model for the integrated switch. In Section 3 we show that the generating functions associated with the data process satisfy a linear matrix equation with the right hand side containing a certain number of unknowns. In Section 4 we show how these unknowns may be determined from finding the roots of an exponential polynomial. These results are then specialized to the single channel case in Section 5, and an analytical formula for the average data delay is derived. As in the exact analysis given in [8], these results are primarily of theoretical interest since the numerical difficulties associated with the solution (except for the single channel case) are formidable. A computationally simple approximate analysis which gives accurate estimates of the average data delay has been obtained and is reported elsewhere [10,14]. Some of these results have appeared previously in report form [10].
R.H. Kwong, A. Leon-Garcia / Integrated hybrid-switched multiplex structure
83
2. Brief description of the queueing model In this section we give a concise description of the queueing model formulated in [4] which we shall use for performance analysis. The movable boundary integration scheme can be modelled as a queueing system with two types of arrivals, voice and data, and an operating rule that allows these customers access to the system at fixed intervals of time (referred to as opening the gate). An arriving voice customer waits in a buffer until the next opening of the gate. If the number of free voice slots is greater than the number of voice customers ahead of him, he receives service. If not, he is lost and leaves the system. For the data traffic, arrivals are buffered, and, at the opening of the gate, are placed on the data slots and any unoccupied voice slots on a first come, first served basis. We assume both the voice and data buffers have infinite capacity, although the finite data buffer case can also be readily handled. The queueing problem to be addressed may now be formulated as follows: given the traffic characteristics, find the blocking probability for voice traffic, and the average delay for data traffic. We adopt the following assumptions for the voice and data processes (the notation is that of [4]). Voice and data customers arrive in two independent Poisson processes with parameters )~ and 0, respectively. The holding time distribution for voice is a negative exponential distribution with mean 1 / # . Data packets are assumed to be of constant size, and a data packet is processed within a frame period, assumed to be of duration b, so that the service distribution function Fd(x), is of the form Fd(X ) = 0 for x < b and Fd(x ) = 1 for x >i b. The numb'er of voice and data slots in a frame is assumed to be N v and No, respectively, with N = N v + N o. We assume that all parameters are expressed in" consistent units.
3. The two-dimensional imbedded Markov chain and conditional generating functions The analysis of the voice traffic performance in terms of the blocking probability has been successfully accomplished in [4], the results of which we quote for completeness later in this section. In order to analyze the data traffic performance, we need to use a two-dimensional imbedded Markov chain to characterize the joint voice and data queueing process. To do so, let us definep v to be the probability of i voice arrivals in a duration of length b, qVj the conditional probability o f j busy voice slots just before the opening of the gate given that there were k present just after the last opening, and pD the probability of i data customer arrivals in a duration of length b. By our assumptions, we have
pV = e xh()~b)'/i!,
i = 0 , 1, 2 . . . . .
q ~ = ( ( ~ )(1-e-~'b)k-j(e-t'b)j' [ 0,
(3.1)
Nv>~k>~J'
(3.2)
otherwise,
pY = e-°b(Ob)'/i!,
i = 0, 1, 2 ....
(3.3)
Let L v be the steady state number of voice customers occupying voice slots just before the gate opens. It is shown in [4] that the voice queueing process is stable when Ob < N - E ( L v)- Furthermore, the steady state distribution of L v, ~rv, i = 0, 1, 2 . . . . . N v is determined by the equations Nv
C = E ¢rVpv,
J = 0,1 . . . . . Nv,
(3.4)
iffi0
and the normalization condition Nv
7rv = 1, j=0
(3.5)
R.H. Kwon& A. Leon - Garcia / Integrated hybrid- switched multiplex structure
84
where pV are the transition probabilities given by Nv--1
z
vv v qkjPk-, + qNvj
k~max(i,j} __
V
--qNvj
pV
for
k~Nv-i
i = 0 , 1 .... ,N v - 1, j = 0,1 . . . . . Nv
f o r i = N v , j = 0 , 1 . . . . . N v.
(3.6)
Finally, the blocking probability PL for voice is given by
kpV+(Nv-j)1-
P L = I - - ~ - - ~ , j=0 7rv[ ~ 1
k=0 ~ pV
+~rVv_l(l_pV)
.
(3.7)
Although, in general, the evaluation of PL requires the solution of the matrix equation (3.4) subject to the normalization condition (3.5), it is shown in [4] that the blocking probability PL is well approximated by the Erlang B formula [11] provided Xb is sufficiently small, which indeed is the case in applications. We now turn to the analysis of data traffic. Let QV and QD be the number of voice and data customers, respectively, in the channel immediately after the t th opening of the gate, with the corresponding steady state variables denoted by Qv and QD- We shall derive the transition probabilities defined by the two-dimensional Markov chain given by the joint process (QV, QD). First, it is easy to write down the following conditional probabilities: min{i,m }
pr{QV=mlQV_l=i,
QtO_l=j } =
~_,
v v qikP,,-k,
m = 0,1 . . . . . N v - 1,
(3.8)
k=0
Pr{QV = NvlQV_l = i, Q~_, = j } =
~o[qVr~v
(3.9)
pv-k ]
and Pr{QD = n I Q v = m,QV_a = i,QD_l = j } =
p~
ifj~< N - i - 1,
P~+N - i - j
i f O <~i + j - N <~n .
(3.10)
From the above equations (3.8)-(3.10) we see that the joint process (QV, Q D) defines a homogeneous two-dimensional Markov chain with transition probabilities Pr(Q v = m, Q,D = n Mo V l = i, 0,°_1 = j ) & P i j , m n given by
=
Pij,mn
~--"
m i n { i, m }
~ v v D qik Pro- k P, k=0 min{i,m} E
k=,
= E k~0
V
V
r
--~N pv-k pD v
qik ~
=
k=O
D
qikPm-kPn+N-i-j
r=N v
Pr-k
J
P2+N-i-j
for
for
j<~N-i-1, m < Nv, 0 ~< i + j -- N ~
for j < ~ N - i - 1 , m =Nv, for
m = N v.
(3.11)
Let the state probabilities of the Markov chain, Pr(Q v = m, Q o = n), be denoted by P ' , . P'n then satisfies the equation t __ P,~, - ~_,pij,,,,P,jt - 1 ,
i,j
(3.12)
R.H. Kwong, A. Leon-Garcia / Integrated hybrid-switched multiplex structure
85
where the summation is over i,j satisfying i,j >1O, i + j <~N+ n. Let P ~ ( z ) = E,~=oPt,z ". We shall call P,,t,(z), m = 0, 1 . . . . . Nv, the conditional generating functions of the data process. F r o m (3.12) we see that
P~,(z) = Y'~ ~ t
P'-lz". ij
Pij,mn
(3.13)
i , j n=O
The right-hand side of (3.13) may be simplified by using (3.11). We find min(i,m} E
Pij,mn zn
=
j ~< N - i - 1,
E
n=0
k=0 min{i,m } =
E
e-Ob(l-z)
V P,, V- ~ e-0b(l qik
k=0
=
V Pm v- k qik
,~v ~
for
-Z)zi+j-N
for
,,v
/'~ik Z.., t,r-k/e-°b~l-z) r=U v
k=O [
=
for
j
qV E PVk e-Ob('-:)Zi+J-N r= Nv J
k=0 [
m < Nv, 0 ~< i + j - N, m < Nv,
j<~N-i-1 m = Nv,
'
for
(3.14)
m = N v.
Substituting (3.14) into (3.13) we obtain, after some computation, that Nv N - i - 1
P~(z) = E i=0
min{i,m}
E
E
j=0
k=O
q,,Vp,.V_ e - e e ( , - z , p , ; - , ( 1 _ z , + j - N )
Nv min( i,m }
+ Y~,
E
i~0
v v e-Ob~l-z)p/- l (z) zi-N q~kPm-k
f o r m = 0 , 1 ..... Nv
1,
(3.15)
k=0
Nv N - i - 1
Puv(Z) = E
i
E
i=0
E qV ~, pVkpij-i e-Ob(1-Z,[l _zi+j-N]
j=0 Nv
r=N v
k=0
i
+ 2
2 qV ~ pV_kpit-l(z)e-Ob¢,-Z)zi-N"
(3.16)
r=N v
i=0 k=0
Using (3.15) and (3.16) it is now easy to characterize the steady state conditional generating functions that will be used later in the evaluation of the average data delay. Let %j denote the steady state probability of having i voice customers a n d j data customers in the system, and let the steady state conditional generating function be denoted by ~r,,(z)= Z,~__0~r,,,,,z", m = 0, 1 . . . . . N v. Assuming that the stability condition for the queueing process holds, we have lim,~ ~ P,,', t -_ ~rm,. On taking limits in (3.15) and (3.16) we finally obtain the following equations for the conditional generating functions ~rm(z): Nv N - i - 1
~r,,,(z)= E i~O
min{i,m}
E
E
j=O
k =0
qikP,,-kv v
e-Ob(a-z)%j(l_zi+y-N )
N v rain{i,m)
+E
E
i~0
vv q~kPm-k e-Ob"-%r~(z)z '-N,
~'Nv(Z) = E
q,,v E
E
i=0
m=0,1,.
, N v - 1,
(3.17)
k=O
j=O
k=O 1.
r=N v
v Pr-k ~ri/ e-0t,~l-z)(1 _ Z,+j-N)
j
Vo[qVvpV_]
+ Y'~ i=0
%(z) e-Ob('-~)zi-U.
=
=
(3.18)
86
R.H. Kwong, A. Leon - Garcia / Integrated hybrid- switched multiplex structure
It will be convenient to re-write (3.17) and (3.18) in matrix form. Define the vector of generating functions ~r(z) T = [%(z), ~h(z) . . . . . ~ruv(z)], where x T denotes the transposed of the vector x. It is clear, by taking terms involving ~r~(z) on the right-hand side to the left, that (3.17) and (3.18) may be written as a vector-matrix equation in 7r(z) as
A(z)~r(z) = b(z),
(3.19)
where A(z) is a known ( N v + 1) × ( N v + 1) matrix consisting of coefficients multiplying the ~r,(z)'s in (3.17) and (3.18), while b(z) is a vector containing the unknowns ~rij, i= 0 ..... N v , j = O, 1..... N - i - 1. To solve (3.19) for rr(z), we must obtain a total of N + ( N - 1)+ . . . + ( N - N v ) = ½(N v + 1)(2N - Nv) N auxiliary equations to determine the unknown ~%'s in b(z). Once ~r(z) is determined, it is then straightforward to calculate the average data delay E(Wo). Thse matters will be dealt with in the next section.
4. Determination of the average data delay In this section we shall derive auxiliary equations for determining the generating function 7r(z), and then use it to compute the average data delay. The method we shall use is based on the following observation. Let Ai(z ) be the matrix obtained from A(z) by replacing the ith column by the vector b(z). Then Cramer's rule says that ~r,(z) --- det Ai(z)/det A(z). But ~r~(z), being a generating function, must be analytic in Izl ~< 1. Hence any zeros of det A(z) in Izl ~ 1 must be cancelled by the corresponding zeros in det At(z ). Such an observation is often invoked in queueing theory [11] and our main task here is to show that this generates the desired number of equations to determine the unknowns in b(z). To do this we first re-write det A(z) in a convenient form for analysis. Let a,g(Z) denote t h e / j t h element of the matrix A(z). Define rain( j,m }
a,,j(z)
=
~-,
qjkVPmV-ke-Ob~l-Z)zj - u
f o r m = 0 , 1 . . . . N v - 1,
k=0 J
-- Y'~ qVk ~ pV k e-Ob'l-Z)Z j - u k~O
for m = N v.
(4.1)
r=N v
We then have the relations
aij(z)---aij(z)
for i ~ j ,
aii(z)=l-aii(z).
(4.2)
From the definition of a~j(z) it is easy to verify that Nv (4.3)
E Otij(Z) = e-Ob(l-z)zJ-N i~O
(cf. (3.8) and (3.9)). Furthermore, in the j t h column of A(z) every element contains a factor of the form z j-N except t h e j j t h entry. From this observation, we may write det A ( z ) = z -tE~'-~°(N-j'ld e t . 4 ( z ) = z - ~ det ,,l(z),
(4.4)
where A(z) is a matrix having as entries
~ij(z)=-~tij(z)
for i . j ,
~.(z)=zU-l--rii(z )
(4.5)
R.H. Kwong, A. Leon - Garcia / Integrated hybrid- switched multiplex structure
87
with min ( i, j }
aij(z)
=
E
qVkpV-ke-ab(1-~)
for i = 0, 1 . . . . . N v - 1,
k=O
J
= y'qVk ~ k=O
pVke-Ob"-~)
for i = N v.
(4.6)
r=N v
Thus the zeros of det A(z) in Izl ~ 1 correspond to those of.~(z) in Izl ~ 1. The number of zeros of A-(z) in Izl < 1 is established by the following proposition.
Proposition 4.1. The number of zeros
o f d e t ,,l(z) in [z~ <~1 is N.
Proof. It is easy to see from (3.8) and (3.9) that Nv
(4.7)
E ~¢ij(z) = e-Ob(l-z)" i=O
Introduce the matrix A(z) whose elements are given by
(4.8) so that
?q,( Z ) = zN -- z' ~,( Z ), 3ij(Z)=--ZJaij(Z),
(4.9)
i4=j.
(4.10)
We may therefore write (4.11) where
- oo(Z)
Z o (Z)
...
zN
O v(Z)
...
zN
l v(z)
B(z) =
(4.12)
aNvo(Z)
ZaNv,(Z)
---
zNva v v(Z)
Note that det .~(z) =
z~J~'Jdet i f ( z ) ,
(4.13)
so that the total number of zeros of det A(z) in Izl ~ 1 is equal to those of det A-(z) plus E~Y~1j zeros at the origin. Now, for Izl ~ 1, an application of Gershgorin's theorem [12] shows that the eigenvalues of B(z), )%(z), i = 0 . . . . . N v, have magnitude I~,,(z)l ~ 1. Thus, from (4.11), we find that if the eigenvalues Xi(z ) are all distinct, the roots of det A(z) are given by solutions of the equations
zU=)~i(z),
i = 0 , 1 . . . . . N v.
(4.14)
An application of Rouch6's theorem to (4.14) shows that for each i = 0, 1 . . . . . N v there are N solutions of (4.14) in Izl < 1. Hence there are a total of ( N v + 1)N zeros of det A(z) in Izl < 1. Thus the total number of
R.H. Kwong, A. Leon- Garcia / Integrated hybrid-switched multiplex structure
88
zeros of A ( z ) in Izl ~< 1 is ( N v + 1)N-Y~, j=lJ uv "= .~, according to (4.13). If the eigenvalues are not all distinct, we can apply the same perturbation argument as in [13] to show that we have the desired number of zeros. This proves the proposition. The following corollary is immediate from (4.4). D
Corollary 4.2. The number of zeros o f d e t A( z ) in tzl <~1 is N. Observe that one of the zeros of det A ( z ) is z = 1. If we denote the other zeros by sp, O = 1 ...... N - 1, we then have the equations d e t A i ( s p ) = 0,
p = 1 . . . . . , N - 1.
(4.15)
These together with the normalization condition ~Irij = 1
(4.16)
i,j
yield the desired number of equations which determine the unknown %j's in b(z). Thus, we may now solve for the conditional generating function tr(z) using (3.19). Once ~r(z) has been determined, the average number of data customers E ( Q D ) is found by Nv
E(QD) = ~ i=0
..1
(4.17)
~zCri(z) z= 1
The average data delay E(WD) is obtained by application of Little's formula [11] to give E ( W D ) = ½b + o E ( Q D ) ,
(4.18)
where the first term in the right-hand side of (4.17) represents the delay due to the gate in the frame structure, and the second term represents the delay once access has been gained to the system. While the above procedure in principle enables us to determine the average data delay, an accurate computation will be a very difficult numerical problem. First of all, A ( z ) is a full matrix with entries being transcendental functions of z, with no special properties that can be exploited. Computing its determinant accurately for moderate values of N v, say around 20, is already known to be a difficult task. After we have evaluated det A(z), which takes the form of an exponential polynomial, we still have to determine its N zeroes in Izl ~< 1. The numerical difficulties involved, even in the case where the exponential factors are absent, have been pointed out by Chang [8]. Our situation would of course be worse in general. Thus the above analysis should be treated primarily as a theoretical contribution to the understanding of the integrated v o i c e / d a t a link model. There is, however, an analytical expression that can be derived for the average data delay in the simplest possible case N v = 1, N D = 0, referred to as the single channel case. This will be presented in the next section. The derivation of this result will also serve to illustrate the procedure discussed above for the computation of the average data delay.
5. The average data delay in the single channel case In this section we specialize the results of Section 4 to the single channel case where N v = 1, N D = 0. This case has also been considered by Weinstein et al. [9] with, however, the crucial difference that they have to assume exponential distributed data packet lengths. The resulting equations in their case are substantially simpler, and involve no exponential terms. The somewhat more intricate calculations required for the determination of average data delay in the constant data packet case will now be presented.
R.H. Kwon& A. Leon-Garcia / Integrated hybrid-switched multiplex structure
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We first write down the equation satisfied by the conditional generating function. Eq. (3.19) for 7r(z) in this case becomes
- ( l - p V ) e - O b . - Z , z -' [ --
l_(l_q'~opV)e-Ob~-z,
pV e-Ob'l-z)~roo(1--z-1) (1 _ p ,) 0o e _ O b , _ Z , ( l _
[~rl(z )
]
z_,)
•
(5.1/
There is only 1 root inside Izl ~< 1, and it is z = 1. We need only the normalization condition Y'.~r, (1)=Evrig = 1 i
(5.2)
i,j
to determine the unknown quantity %o in (5.1). Solving for %(z) and %(z) we obtain, after some computations, % ( z ) = p v e - o b ' l - ~ ' % o ( z - 1 ) [ 1 - q v l e oh,,-~)]
~v _v~~ 20h(1-~) } × ( z [ 1 - e - ° b ( 1 - z ) ] - p v e - O h ( l - z ) + zqVop v e -°ml-z) +'tllt'o
1
,
(5.3)
¢rl( g ) ='n'oo e-°b(l-z'( g - 1)(1 _pV)
× (z[1-e-°ba-z']--pVe-°b"-z)+zqVopVe-°b'l-z'
+ qupoV v e - 2 0 b . - = ) ) - I
(5.4)
Since the numerators and denominators of %(z) and ~h(z) all vanish at z = 1, we need to apply L'Hospital's rule to evaluate %(1) and %(1). We find
v v v v + qa, v Pov Ob - Ob), % (1) = Poqm%o/(qmPo V
V
V
% (1) = %o (1 - Po )/(qloPo + qVl P vOb - Ob).
(5.5) (5.6)
Using the normalization condition (5.2) we get
%o = ( qlV p~ + qVlpVOb - Ob)/(1 - pV + qVopV ).
(5.7)
Eqs. (5.3), (5.4) and (5.7) completely determine % (z) and % (z). Now the generating function for the steady state data customers QD is given by
a~(z) =
E~,(z) i
so that in this case
Go( z ) = %o e-°b'l-~)( z - 1)[1 -PoqnV v e-O/,(l-~)] X{z[l_e_Obo_z)]_pV
e_Obd_~)+zqVopV e_Ob(l_~,+qVlpV e_2Obo_~) } 1.
(5.8)
The average number of data customers is then given by d E(QD) = ~zz GD(Z)~=1"
(5.9)
Let u(z) and v(z) be the numerator and denominator of GD(Z), respectively. G'D(z) will be indeterminate at z = 1 since its numerator as well as denominator vanish at z = 1. Applying L'Hospital's rule to G~( z ) gives G;(1) = 0'(1) u"(1) - u'(1) v"(1)
2[v,(1)] 2
(5.10)
90
R.H. Kwon& A. Leon-Garcia / Integrated hybrid-switched multiplex structure
2800
2400
2000
o 0) m
1600
m
1200
800
400
10
20
30
40
SO G (packets/sec)
Fig. 1. Average data delay as function of data load (constant data packets): N v = 1, N D = 0, ~ = 0.01, # = 0.01, b = 0.01.
Computation then yields EQD = [2(qV0pV + qVlpVOb - Ob) %o8b(1 - 2qVlp v) - %o (1 - pVqV1) Ob(2qVo pV + 3qVl pVOb _ Ob - 2)][ 2( qV1pVOb + qlv pV _ 85 )2] -1 (5.11) Finally, the average data delay is given by (4.18). As an example, for X = 0.01, /~ = 0.01, b = 0.01, the average data delay is plotted against 0 in Fig. 1. Notice that from the approximation using the Erlang B formula, E ( L v ) = 0 . 5 so that the stability condition for the data queue is 0b < 0.5. This is shown clearly in Fig. 1. When compared with the results of Weinstein et al. [9] for the single channel case using the assumption of exponentially distributed packet lengths rather then constant packets lengths considered here, we see that the results are very close, as would be expected. 6. Conclusions
We have presented an exact analysis of the integrated hybrid-switched voice/data multiplex structure previously considered by Fischer and Harris, using the method of two-dimensional imbedded Markov chains and conditional generating functions• The determination of the average data delay requires finding
R.H. Kwong, A. Leon -Garcia / Integrated hybrid-switched multiplex structure
91
the roots of an analytic f u n c t i o n of a complex variable within the u n i t disc, a n d the solution of a v e c t o r - m a t r i x equation. While the p e r f o r m a n c e of the integrated switch m a y thus in principle be evaluated, it is evident from the solution procedure that the required c o m p u t a t i o n s will be too complex to carry out in practice except in extremely simple cases. The results are therefore primarily of a theoretical n a t u r e a n d a suitable a p p r o x i m a t e p e r f o r m a n c e analysis is needed to u n d e r s t a n d the traffic b e h a v i o u r in realistic situations. O n e such a p p r o x i m a t i o n based o n a fluid a p p r o x i m a t i o n technique, which is c o m p u t a t i o n a l l y attractive a n d whose accuracy has been verified b y simulations, has been reported in [10,14].
References [1] I. Gitman and H. Frank, Economic analysis of integrated voice and data networks: A case study, Proc. IEEE 66 (1979) 1549-1570. [2] G. Coviello and P. Vena, Integration of circuit/packet switching by a SENET (Slotted Envelope Network) concept, Proc. National Telecommunications Conf., New Orleans, LA (1975) 42.12-42.17. [3] K. Kummerle, Multiplexer performance for integrated line and packet-switched traffic, Proc. 2nd Internat. Conf. on Computer Communications, Stockholm, Sweden (1974) 517-523. [4] M.J. Fischer and T.C. Harris, A model for evaluating the performance of an integrated circuit- and packet-switched multiplex structure, IEEE Trans. Commun. COM-24 (1976) 195-202. [5] B. Occhiogrosso, I. Gitman, W. Hsieh and H. Frank, Performance analysis of integrated switching communications systems, Proc. National Telecommunication Conf., Los Angeles, CA (1977) 12.4.1-12.4.13. [6] E. Arthurs and B.W. Stuck, A theoretical traffic performance analysis of an integrated voice-data virtual circuit packet switch, IEEE Trans. Commun. COM-27 (1979) 1104-1111. [7] E.A. Harrington, Voice/data integration using circuit
switched networks, IEEE Trans. Commun. COM-28 (1980) 781-793. [8] L.H. Chang, Analysis of integrated voice and data communication network, Ph.D. Thesis, Dept. of Electrical Engineering, Carnegie-Mellon Univ., 1977. [9] C.J. Weinstein, M.L. Malpass and M.J. Fischer, Data traffic performance of an integrated circuit- and packetswitched multiplex structure, IEEE Trans. Commun. COM-28 (1980) 873-878. [10] R.H. Kwong, A. Leon-Garcia and A.N. Venetsanopoulos, A study of a joint stochastic process for an integrated network, Final report, Prepared for the Department of Communications, Canada, under Contract OSU79-00041, 1980. [11] L. Kleinrock, Queueing Systems, Vol. I (Wiley, New York, 1975). [12] G. Strang, Linear Algebra and Its Applications (Academic Press, New York, 1976). [13] M.F. Neuts, A queue subject to extraneous phase changes, Adv. Appl. Probab. 3 (1971) 78-119. [14] A. Leon-Garcia, R.H. Kwong and G. Williams, Performance evaluation methods for an integrated voice/data link, IEEE Trans. Commun. COM-30 (1982) 1848-1858.