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ELSEVIER Computer Networks and ISDN Systems 29 (1997) 1919-1931

Providing the statistical QoS objectives in high-speed networks

Hoon Lee a, * , Yoshiaki Nemoto b a Switching Technology Research Laboratories, Korea Telecom, 17, Umyon-Dong, Socho-Gu, Seotd 137-792, South Korea

b Graduate SchooI of Information Sciences. Tohoku University, Aoba, Aoba-ku, Sendai 980, Japan

Abstract

We present an approximate analytic model for providing the statistical QoS (Quality-of-Service) objectives in High-Speed Networks. First, we derive approximate formulae for the tail distribution of the queue length and the cell delay for the discrete time G/D/c queueing system with correlated input arrivals. Next, we present the relationship between the tail distribution of the queue length with the required service rate in the statistical aspects. Next, we present the tail probability of loss period ar a statistical QoS measure. We assume the statistical multiplexer of the ATM output port with cell admission regulations based on the arrival processes, and investigate the performances. Finally, we verify the accuracy of the theory via experiments of the numerical computations and simulations. 0 1997 Elsevier Science B.V.

Keywords: Statistical Quality-of-Service; Tail distribution; Delay loss; Delay loss period; ATM output multiplexer

1. Introduction

The typical Quality-of-Service (QoS) measures for the services in ATM networks are the cell loss and delay. For the applications having real time constraints, the cell delay such as the maximum tolerable delay to transmit a cell across the network plays more important role. In other words, if a deadline is associated with each cell and if the it is supposed to reach its destination after its deadline is passed, it ma.y be considered useless and is subject to be discarded a priori [1,2 I]. We denote this loss as the delay 10s.~.

Typically, the QoS values are specified on an end-to-end basis. However, the exact solution to the end-to-end delay is hard to obtain. Thus, the appor-

* Corresponding author. E-mail: hlee@traffic.kotel.co.kr.

tionment of the end-to-end QoS values to the local QoS values in a node is usually adopted [21]. So, in this paper, let us confine our interests in an element- wise delay loss in a switch, and determine that an arriving cell is discarded if it finds the queue length to be the greater than the predefined value.

As for the guarantee of the delay QoS, there exist two important problems that should be addressed. One is the input regulation for the gracious degrada- tion of the delay loss. This has a significant effect on the performance of the overload-prone case, espe- cially in the ATM multiplexer with high load which accepts highly bursty arrivals 1181. Because, for the real time services such as voice and video, if the system has a possibility to fall into heavily loaded state, some portion of the cells can be discarded without sacrifices in the QoS felt by a user. By doing that, the QoS will degrade smoothly (thus, gra-

0169-7552/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO169-7S52(97)00103-7

1920 H. Lee, Y. Nemoto/ Computer Networks und ISDN Systems 29 (I 9971 I91 9-1931

ciously), otherwise heavy degradation of the QoS may occur by elongated overflow states. Another is the measure that how long the hazard state such as the delay loss period lasts [ 18,191. The measure of loss period length has another important meaning in the system performances if the interests lie in the time correlated real time cell streams.

It is known that there are two major types for the QoS guarantee: the deterministic guarantee and the statistical guarantee [7]. Among them, the statistical QoS guarantee is considered to be more suitable in investigating the probabilistical behavior of the QoS and is recognized as more realistic method for solv- ing the above mentioned problems.

Therefore, in this paper, the statistical QoS mea- sure is considered and it is defined as follows. The probability that the delay experienced by a cell is larger than a specified target value does not exceed some value required by a service. That is, the statisti- cal QoS measure is formulated as follows [7]:

Prob(delay 2 target value) I bound.

The researches into the tail distribution of the queue can be found from several literature [5,8, 14,15,25]. On the other hand, the following literature deal with the statistical QoS measures for the ATM networks. Ferrari [7] and Kurose [16] defined the general concept of the statistical QoS guarantees. Roberts and Virtamo [23] proposed upper and lower bounds for the queue length distributions for the superposed periodic cell arrivals via non-exponential approximation functions.

Hui [12] had applied the large deviation tech- niques to evaluate the probability of burst blocking in the broadband networks. Kesidis et al. [13] de- rived the approximate formulae of the queue length distribution for the Markov fluid source and the Markov modulated sources via heuristic method. Guerin et al. [9] made comprehensive discussions for the approximate analysis to the equivalent capacity for the networks, whose concept is similar to the statistical QoS measures. Chang [3] proposed a math- ematical model for the deterministic upper bound for the tail probability of the queue length and delay of the G/D/c queue in discrete-time and discussed their stability condition for the cell arrival process called the envelope process. Yaron and Sidi [26] proposed a model for the upper bound of the queue

length using exponentially bounded arrival process by imposing exponential decay in the distribution of the burst length. However, Yaron and Sidi’s model assumed the coefficient of the exponents to be con- stant with respect the decay rate, and there is no explicit formula for the decay rate.

There are many works which deal with the cell acceptance control for the statistical multiplexer in view of the mean performances 16,171. However, if it is concerned with the statistical QoS measure, we could not find any work yet to our best knowledge. Thus, in this paper, we discuss the statistical QoS guarantee problems by modelling analytically the tail probabilities for the cell delay loss and its period, and investigate its performance by considering the cell acceptance control. First, based on the concept of the large deviation techniques, we derive an ex- plicit closed-form formula for the upper bound of the tail distribution of the queue length and delay for the general dependent arrivals and deterministic service system. Next, we will apply those results to the problem of statistical QoS guarantee which takes into account the cell acceptance control for the out- put multiplexer of the ATM networks.

The remainder of this paper is composed as fol- lows. In Section 2, we present some preliminaries concerning approximate upper bounds for the tail distribution of the queue length for the general G/D/c queue with correlated arrival, from which we derive the upper bound for the tail distribution of the cell delay. In Section 3, we give the formulae for the statistical QoS measures such as the mean ser- vice rate which guarantees the statistical cell delay requirements, and the tail probability of the delay loss period by applying the results of Section 2. In Section 4, we present the numerical results via com- putations and simulations and, give some discussion. Finally, in Section 5, we summarize the paper.

2. Preliminaries

2.1. Queue model

Let us consider a discrete-time G/D/c queue which is a typical queue model in ATM context.

H. Lee, Y. Nemoto / Computer Networks und 1SDN Systems 29 (I 997) 1919-I 931 1921

Suppose that the time is divided into slots of unit length during which a burst of cells arrives, and at most c cells depart the queue just before the end of time slot. The cell egress follows the FIFO principle. For the cells anrived simultaneously in a slot, a cell egresses in random order. Let us assume that the arrival process igenerates a batch which is dependent on the state of the source, and it is correlated with respect to the time slot. The arrival process is gov- erned by a modulating process G = {Gk. k = 1, 2 , . . .}, which we call a phase process. G, is as- sumed to be a Markov chain on the state space

A={l,2, . ..) IM} with the transition matrix P has an element pij = Pr{Gk+ , = jlG, = il. Let A,(G) =

i A’,, i&H’, k:= 1,2, . . . } be a sequence of random variables, which describes the number of cells that arrive during time slot k when the source’s phase is G, = i for A;.

Now let us assume that the input regulation is carried out as follows: The input controller regulates the ingress of the cells generated by a source based on the phase of the source, and it is governed by the following function: Yl = f( A;), that is, among the A’, cells arrived in a slot k when the modulating process is in phase i, only Yk cells can be admitted into the queue, while the remaining (A; - Y,‘> cells are discarded.

Let the random variable Xi be the queue length at the beginning of time slot k given that the phase is i at that time slot. The system state evolves at the boundaries of the time slot. Note that, in the sequel, the superscript indicates the phase of the arrival process, which is omitted without loss of generality when it is possible, while the subscript indicates the time slot index and its related representation. We assume that X, = 0, that is, the system is empty at the initiation of a slot.

Then, we Ican represent the number of cells re- mained in the queue at the beginning of time slot k,

X,7 ‘v X,=Max[X,-, + Yk-, -c,O]. (‘1 Fig. 1 illustrates the queue evolution between the time slots.

2.2. Upper bound for tail distribution

In order to obtain the upper bound of the probabil- ity that the cell delay exceeds a certain value, we

yk-l yr*7 yk i {{ .,, 4 + I ’ (k-l)-th slot

b

k-th slot ’ -) Time

C 4

Fig. I. Queue evolution between the time slots.

first have to derive the formula for the upper bound of the queue occupancy distribution. As an approxi- mate method for the upper bound of the queue occupancy, the large deviation techniques are consid- ered to be a useful method [2,41.

Referring to Eq. (I), let us denote the variation of the queue length in a time slot k to be 2, = Yk - c. We also denote a function for the cumulative varia- tions between time slot [k,l] as Z,,, = C:,,Z,, and let L,,,(B) = (l/m)logE[exp(OZ, ,I], where m = I -k+ l,m>O. We denote a limit by L(O)= lim m-r~ L,(8), t/8 E R, where R is a positive real set. L(B) is assumed to be strictly convex ’ 1241 and differentiable for 0 < fI < 3~. The convexity of the function L( 0) can be easily proved by using Halder’s inequality (101.

Then, for E > 0, we can assume that there exists mE such that for all m > m,

E[exd%.d] ~ex~[V0) + +I. (2) From Eq. (1) we obtain X, = Max,,,-,+,Z,~,, and if we use the inequality Max(x, y) I x + y, for x,y 2 0, then we obtain

E[exd WJ] 2 i E[ev( f%..J] m=l

m-1

+ C ed(L(@) +E)ml. (3) m.m, If I,( 0) < - E holds, then we have

E[exp( ex,)] s C( e> < r,

I g(.) is said to be convex if and only if ~(a.x +(I - a)~)<

q(x)+(l-- u)g(y) for alI x.y E I, I is a set of integers, and all

UE[O,ll.

1922 H. Lee, Y. Nemoto / Computer Networks and ISDN Systems 29 (I 997) 1919-I 931

where

C( 0) = ? E[expWdl. (4) m=l

In order to estimate the upper bound of C(e), let f&,,(e) = E[exp( eZ,,,>l, and define n(e,m) = sup,(l/0)log Q,,(e). If there exists A(B) which satisfies A(B) < c, and an arbitrary small real valued s(e) such that n(e,m> = (A(B) - c)m + S(e), then we have

w - c = lim Q(e,m) -s(e) W)

= sup - m-x m e 8 ’

and

s(e)= SUP su&logn,(ej -L(e)m) . (5) [ mz1 e 1

In order to obtain a closed formula for c(e), we take the summation over = in formula (41, which results in a rough upper bound. That is,

c(e)5 i exp[e((A(e)-++6(e))] m=l

= exp[e(A(e) --c+ %@))I 1 -exp[e(h(e) -c)] ’ (6)

Finally, from the Markov inequality [ 111, we have

Pr( X, 2 x} = Pr{exp( ex,) 2 exp( 8x))

E[exp( ex,>l I

exp( Ox) ’

from which we obtain

pr{x,2~} sc(e)exp(-Ox), (7)

where 8 is the largest solution of the inequality L(d) < 0. We call 0 the decay rate and C(0) the prefactor.

Inequality (7) holds for the stationary distribution as well if the sequence {Z,,} is stationary and ergodic. We prove this in Appendix A.

Now let us consider the tail probability of the delay. We define the delay as the number of time slots an arriving cell has to wait till it receives the service. For the FIFO server, the delay depends on the number of the cells waiting for the service in the queue. For the case of a sufficiently large queue

length x, this can be induced from the result of formula (7), since, in that case, the server serves c cells every time slot almost surely. Thus, for the case of sufficiently large x (x B- c> and t, the delay limit imposed upon the cell, we can consider that the tail probability of the delay experienced by a cell has a mean with the same distribution as that of queue length.

We will prove this fact. From Eq. (I), we can deduce that, as we had assumed the randomness among the cells that arrived simultaneously in a slot, the minimum delay that is experienced by a cell that arrived during the time slot k (we call it just a delay henceforth) is upper-bounded by [ XJcl, where [ yl indicates the smallest integer greater than or equal to y. That is, the tail probability that the cell delay exceeds t time slots is equivalent to the probability that the queue length exceeds x, where t = [x/cl. So, we obtain the following upper bound:

Pr i 1

+2t IPr(X,2c(d- 1)).

Thus, if we denote W, to be the minimum delay of a cell which arrives during time slot k, then we have the tail probability for the cell delay given by

Pr(W,lt] sc(e)exp(--ce(t-- 1)). (9)

2.3. Decay rate

In order to obtain a formula for the decay rate of the approximate formula (7), let us consider the detailed behavior of the admitted arrivals. Without loss of generality, we call the admitted arrivals just arrivals. We define the moment generating function (mgf) of the arrival process by 9 ‘( 0) = E[exp(B Y,i)].

Let us denote a diagonal matrix U(0) by q(e) = diag(ll”(e), q*(e), . . . , lu”(e)), where diagc.1 indicates the matrix composed of diagonal compo- nents given by (.> while the other elements are equal to zero. Let a matrix F = ly(0)P. We denote g(F) to be the spectrum of F. The spectrum of F is defined to be the set of all r E R that are eigenvalues of matrix F. Let sr(F) be the spectral radius of the matrix F defined by sr(F) = max{lrl: T E (T(F)}.

In order to derive the relationship equivalent to the stability condition, let us define a limit function

H. Lee. Y. Nemoto/ Computer Nehvorks und ISDN Systems 29 (1997) 19/9-1931 1923

by M(B) = lim,,, (1 /m) log E[exp( 8 Yk,,)], where Y,,, = C{=,Y,. Then, we have two results with re- spect to M(8), which is given as follows:

1. M(O) is upper-bounded such that M(8) I log sr(F).

Proof. For the arrival process with generator G. we have the following formula.

E[exr@ Y, .,J]

(10)

where Y,,,+ is the number of arrivals at the end of time slot m given that the Markov chain was in state i at the initial time slot, and pi, indicates the transi- tion probability of the Markov chain from state i to state j.

Let @,,(B > = (E[exp{O Y,,,,,,)l, . . ,E[expl~ Y,.+, )]) and let @,,,(0)T be its trans- pose. If we write Eq. (10) in matrix form, it is given

by

@,&3)T=F@:,_,(B)T. (1’)

An initial condition is given by @,(OjT = !?!(8)IT, where I is a raw element matrix. Now let 7~~ be the initial probability that the Markov chain is in state i, and its vector is given by x = (7~,, . . ,nTTM). Then, from the property of the conditional probability, we have

E[exp{eY,.,:t] = ~@k(fvTT

which in turn becomes

E[exp{BY,,,]] = ~(F)“-‘~(0)IT. (12)

From the property of the convergence of the matrix norms with respect to the spectral radius [lo], the matrix (FY” IS upper-bounded by p(BXsr(F) + elm, where p(0) and E are arbitrary positive constants. This is proved in Appendix B. Then, from Eq. (12). the assertion 1 is proved. Cl

2. If the Markov chain is irreducible and aperiodic, then the equality holds, that is, M(8) = logsr(F).

Proof. Since we had assumed the ergodicity of the Markov chain, the transition matrix P is primitive,

that is, P” > 0 for n 2 1. As a result, the matrix F is primitive because V’(0) is positive. In order to de- rive the final result, we introduce the Perron- Frobenius theorem for the eigenvalue and its eigen- vector, which is summarized in Appendix C.

Thus, based on this theorem and from Eq. (12), the assertion 2 is proved. 0

Finally, we can obtain the decay rate from the stability condition of M(B) < 4. That is, we have 0 which is a largest solution of the inequality

fl> ilugsr(F). (13)

3. Statistical QoS measures

By using the result of Section 2, we consider, the statistical QoS (SQoS) measure for an ATM switch.

3.1. SQoS and required service rute

Let us consider the relationship between the SQOS required by a service and the corresponding system capacity. Given the arrival processes and the SQoS constraints defined by the upper bound on the proba- bility distribution of cell delay, we estimate the mean required service rate. We denote the effective service rate as the mean required capacity of the server in equilibrium for the output multiplexer of a switch to fulfill the required SQoS. The notion of effective service rate is similar to that of the effective band- width which is defined in [9]. The effective band- width is the minimum mean service rate required by a system to guarantee a QoS, e.g., the buffer over- flow probability, defined by an aggregation of sources.

Then, for the deterministic server we had as- sumed, the SQoS for the delay loss t is mapped to the probability that the queue length X, where X = et, of an infinite capacity queue exceeds a specified value K should be below a certain value K. Alterna- tively,

Pr(XtK} SK, (14)

1924 H. Lee, Y. Nemoto / Computer Networks and ISDN Systems 29 (I 997) I91 9-1931

where K is an integer and K is a real number. Thus, if we relate the inequality (14) with the inequality (71, we obtain

Pr{X>K}<C(8)e-Ke=K. (15)

Then, we obtain the following equation

(16)

which can be solved by numerical root finding method.

3.2. A distribution for delay loss period

Based on the definition of SQoS measure, let us assume that an arriving cell is lost when it finds that the queue length is greater than or equal to a thresh- old value K. This implies that the arriving cell can not be guaranteed its delay requirement if the queue length is not smaller than K. Thus, it is lost a priori. We call the period during which the queue length is greater than or equal to a threshold value K as a delay loss period. In order to obtain a distribution for a delay loss period (DLP), let us investigate the evolution of the queue length with respect to time slot, which is illustrated in Fig. 2.

Let us divide the DLP into two time periods; one is the period composed of time slots that have passed already at the observation point, the time slot k, and the other is the period composed of the time slots that may last from that instant. We call each of them the trace and the residue, and is denoted by h, and r,, respectively. Let us first consider the trace. The trace of DLP observed at time slot k is described as follows:

h,=Min[j:X,>KIX,-,-=K]. (17)

We approximate that the following upper bound holds:

Pr{ h, =j} I Pr{Xkmj = K)Pr(Yk-j.k ljc}- (18)

Then, the probability that the length of the trace is not smaller than 1 is given by

Pr{h,rl) = iPr(h,=j}. (19) j=i

Time Slot

’ Delay Loss Perio

Fig. 2. Sample path of delay loss period.

Let us rewrite the formula (18) into matrix form by taking into account the phase of the source at the observation point. In order to do that, let us define some variables. Let h,(j) = (hi(j) hi(j) . . . hr( j)) be the matrix composed of the probability h;(j) = Pr{ h, = jlsource phase = i). Let us write the condi- tional probability of queue occupancy which is de- fined explicitly in Appendix D as follows:

Xk-j( K) = ( Xi-,( K), . . . TX,“j( K))T. (20)

Let us denote E= diag(w, w, . . . , w), where w = Nyk-j,k 2 jd.

Then from (18) and (20), we obtain

x,(j) I ,ykmj( K)TE. w

where x~- j( I( jT is the transpose of the matrix x/I _ jfK). Note that, in Appendix D, we assumed that as the queue length is not smaller than the threshold value K, the arriving cells are assumed to be re- jected, which gives us an equivalent finite state queueing system with state variable ranging {0,*, . . . , K}. Note that this assumption is restricted only to the derivation of Eq. (34), otherwise we relax this assumption. Finally, the probability that the trace is not smaller than 1 is given by

i,( 1) = i&(j) I ~ X~-j( K>‘a. (22) j-l i-1

Now let us consider the residue of DLP. The residue of DLP seen by an arbitrary observation point k is given as follows:

r,=Min[i: X,,, <KIX,zK]. (23)

H. Lee, Y. Nemoto / Computer Networks and ISDN Systems 29 f 1997) I91 9-1931 I925

Since the residue is conditioned on the trace, we have

Pr{r,li)= i,Pr{(h,=j)fl(r,zi)}. (24) j=O

If we use an independence approximation between the two events, we obtain

Pr{r,li} I f,X,-JKjTZZ*, j=O

(25)

where E*=diag(w*,o*, . . ..w*) and o* = Pr{Y,,,+i 2 ic}. Let us rewrite the formula (25) into

Tk(i)l ~x~-~(K)~SS*, (26) j=O

where F,(i) = (r:(i), r:(i), . . . , rkM(i)). The DLP, which is observed at time slot k and will iast 1 time slots in total is. given by DLP, = h, + rk = f, and its tail probability is given by

Pr{ DLP, 2 I} := Pr{ h, + r, 2 I}

(27)

Let us denote m,(I) as m,(I) = (DLPL(I), DLPl(/>, _ . _ , DLP:(/)), where DLP$I) = Pr{DLP, 2 [(G, = i). Then, we have an approximate formula for the upper bound of the probability that the delay loss period is not smaller than 1 time slots given by

where 8= diag(3, ij, . . . , B) and 3= Pr{y,-j 2 (I - j)c}, where y, indicates the mean cumulative arrivals over 1 time slots, which means the mean of the total number of cells arrived during the I time slots. Now let us consider the approximation of the upper bound for the stationary probability of the loss period. We do this heuristically by observing the formulas (7) and (181, from which we obtain the stationary upper bound for the loss period as follows:

Pr(DLP>t) rPr{X=K}Pr(y,2Ic}, (29)

where the explicit formula for the first term in the right hand side is obtained by mathematical induc- tion between the probability mass function and its tail distribution, to be Pr{X = K) = C(0Xl - ewe)ewBK.

4. Performance evaluation

We present the results of the performance con- cerning the SQoS measures via the numerical com- putation and the simulation experiments. We assume various source cases; the binomially distributed source as an example of time independent bursty source, and the On/Off source as a time correlated two-phase source, and the three-phase Markov mod- ulate source as that of more complex time correlated source.

First, we give the upper bounds for the tail distri- bution of the cell delay loss probability. Next, we present the estimation of the mean service rate re- quired to guarantee the delay loss SQoS. Finally, we illustrate the results for the loss period distribution.

4.1. Source model

First, let us assume a three-phase source as shown in Fig. 3. If we describe its phase transition in matrix form, it is given by

[

1-a-p ff P P= 6 1-6-y Y

I

. (30) rl 5 l-77-5

We assume that, in phase i, i = 1, 2, 3, the source generates cells with arbitrary rate function r,.

c-l 1-c-p 1

1-E-r co#o% Y 11

2 -3 03 -5

I-0-l

Fig. 3. State transition diagram for three-phase source.

1926 H. Lee, Y. Nemo~o / Computer Networks und ISDN Systems 29 (1997) I91 9-1931

Second, let us consider an On/Off source. The state transition matrix of On/Off process would be given as follows:

P= [

1 - 92 q2

91 I l-9,. (3’)

The modulating chain has two states, state 1 and 2, and cells arrive with rate r, and r2 for each state, respectively. In each time slot the phase will changes from state 2 (or 1) to state 1 (or 2) with probability 9t (or q2). Otherwise, the phase remains in state 2 (or 1) with probability 1 - 4, (or I - q2).

Finally, let us assume an arrival process which has just a single phase; and the cell arrivals are rene’wal with probability p [20].

Returning to the three-phase source, let us give the regulation function for k > 1 as foliows:

rl if G,= 1,

Yk = v2 if G, = 2, (32)

5r3 if G, = 3.

The diagonal matrix of the moment generating func- tion for the arrival process is ‘I’ = diag(e”l, ew> , ee5’l).

4.2. Tail probability of cell delay loss

Let us first consider the renewal arrival: An ag- gregation of N connections constiiutes a batch of cell arrivals, and we assume that the number of cell arrivals in a slot is distributed binomially: That is, the parameters for the cell arrivals are given by a set (N, p) and N, the number of associated connections, is assumed to be N = 20. The probability p has three values 0.045, 0.040, and 0.035 which correspond to the offered load of 0.9, 0.8 and 0.7, respectively, when the service rate c = 1. We denote the three parameter sets as P 1 = (20,0.045), P2 = (20,0.040), and P3 = (20,0.035), respectively.

Using these parameters and from the formula (13), we obtained the decay rate, ~9, given by 0.233, 0.521, and 0.888 for each parameter set PI, P2, and P3, respectively. In order to obtain the decay rate with less computational complexity, we approxi- mated the binomial distribution into the Gaussian one [20] and used the Legendre transform, from which we calculated the decay rate.

‘e-2010 20 30 40 50 60 70 60 90 H)

t, Time Slot

Fig. 4. Delay loss for binomial source.

Fig. 4 illustrates the upper bound for the tail distribution of the cell delay loss probability. The real lines indicate the results obtained by the numeri- cal computations, and the marked points indicate those obtained by simulations. This notation is ap- plied to all the cases in the sequel unless explicitly noted. From the results, we can find that the approxi- mate analytical results are in good agreement with the simulational results.

As for the On/Off source, we can obtain an explicit formula for the eigenvalue from the quadratic equation, which is well known (for example, refer to DOI).

We assumed two cases of parameter set E = (9,,q2, r,, r2): that is, El =(0.5,0.4,0,2) and E2 = (0.65,0.35,0,2). The service rate is assumed to be one cell in a slot. For these parameter sets El and E2, the mean cell arrival rates ,are obtained to be 0.8894 and,0.699, and the decay rates are obtained to be 0.1823 and 0.6190, respectively. Fig. 5 illustrates the results for the tail distribution of the cell delay loss of On/Off source.

t, Time Slot

Fig. 5. delay 16~s for On/Off source.

H. Lee, Y. Nemoto/ Computer Networks and ISDN Systems 29 (19971 I91 9-1931 1921

le-104 t- 6 8 10 12 14 16 16 20 f

t, Time Slot

Fig. 6. Delay loss for three-phase source.

Finally, as to the Markov modulated three phase source with a parameter set T= (a, /?, 6, y, q, 0, we assume two cases: a source with relatively even phase transition (Tl) and a source with elevating variation between the states (T2): Tl = (O&0.3, 0.3,0.4,0.5,0.3) and T2 = (0.6,0.1,0.2,0.7,0.6,0.2). The arrival parameter has a set ( r, , r2, r3> = (1,2,3). The service rate is assumed to be c = 2.

The stationary distribution for the two sets, Tl and T2, were obtained to be (0.328,0.366,0.306) and (0.365,0.315,0.321), respectively. The mean ar- rival rate for the T 1 and T2 are given- by (1.980, 1.956). The decay rate for each parameter set Tl and T2 are obtained to be 0.1091 and 0.1978, respectively. Note that, for the case with phase size greater than two, there is no explicit formula for the eigenvalue of matrix F defined in Section 2.3. It is obtained only by numerical root finding method. Fig. 6 illustrates the results for the upper bound of the cell delay loss rate for the parameter set T2. As we

2r

a, 1.6 .o 2 g 1.4

.$ 1.2

H

E '

0.8

1. '% 20 30 40 50 60 70 80 90 100

Threshold K

1.5' I 10 20 30 40 50 60 70 60 90 loo

Threshold K

Fig. 7. Mean required set-vice rate for binomial source Fig. 9. Mean required service rate for three-phase source

I I 10 20 30 40 50 60 70 80 90 100

Threshold K

Fig. 8. Mean required service rate for On/Off source

can see from the result as well as in the previous two experiments, the approximate upper bound has a good agreement compared with the time consuming simulation result. This indicates that the proposed formula can estimate well the cell delay loss proba- bility with simple and less expensive computational costs compared with the time consuming standard simulation.

4.3. Required service rate

Now let us compute the mean service rate re- quired by a server in an output multiplexer to satisfy the statistical QoS requirements. We assumed the same three arrivals as in Section 4.2. Here, the parameters defined in the previous subsection will be assumed for each source unless it is newly defined.

First, let us consider the binomially distributed source. The target SQoS requirements for the cell delay loss are assumed to range from 10e6 to 10d9.

Target S-QoS= 16’ for all K

1928 H. Lee, Y. Nemoto / Cvmputer Networks and ISDN Sysrems 29 (I 997) I91 9-1931

zi I- le-07 2 3 4 5

I, Loss Period

I- lem06 - 2 3 4 5

I, Loss Period

Fig. 10. Probability of loss period for binomial source Fig. 12. Probability of loss period for three-phase source.

Fig. 7 illustrates the mean service rate for system with the binomially distributed source. The threshold is assumed to vary from 10 to 100 and the upper group, the group P 1, of the curves indicates the case when the average cell arrival rate with p = 0.045, while the lower one, the group P3, indicates that of p = 0.035. We found that the required service rate for more severe QoS application is greater than that of the less severe one. For example, for the queue size of 10, the required service rate is 2 times greater than its mean arrival rate. This becomes smaller as the queue length becomes larger.

Second, let us consider the On/Off source. The target SQoS requirements are assumed to range from 10m6 to 10e9. Fig. 8 illustrates the result. The upper group is the case of El, whereas the lower group is the case of E2. For the same queue size and the average cell arrival rate, we can find very similar trend in the mean service rate.

Finally, let us consider the three phase source. We assume the following parameter set for the access

5 0.1

a” K=lO K=lO 1

I s

KG0 -

2 2 3 3 4 4 5 5 I. Loss Period

Fig. 11. Probability of loss period for On/Off source.

regulation: (cp, 6 I= (O&0.7). While, the target SQoS requirement is assumed to be IO-‘. Fig. 9 illustrates the result for the required service rate. Note that the effect of the input regulation becomes noticeable as the threshold becomes smaller.

4.4. Probability of delay loss period

We investigate the time period the queue is in lossy state. The parameters used are the same as in the previous section unless it is defined explicitly.

Fig. 10 illustrates the tail probability distribution for the loss period when the input is a binomially distributed source. The lines indicate the analytical results, whereas the points indicates those obtained by simulational experiments. In Fig. 10, we assumed that the mean offered load of 0.9, which corresponds to the superposed independent sources with the pa- rameter (20,0.045). The threshold is assumed to be K = 10,20, and 30. The arriving cells are discarded if they find that the queue length is not smaller than the value K when he arrived to a queue. The abscissa indicates the consecutive time slot of loss

period varying from 2 to 5. We found that the greater the threshold is, the closer the approximation toward the simulation results are, which we had expected.

Fig. 11 illustrates the results for the G/D/I queue with the On/Off source as an input. From the results, we knew that the two results (obtained by computation and simulation) have a good agreement for the case of K = 20 and K = 30, whereas it overestimates when the queue length is K = 10 which is considered to be small.

H. Lee. Y. Nemoto / Computer Networks and ISDN Systems 29 (1997) 1919-I 931 1929

Fig. 12 illustrates the results for the G/D/2 queue with the three-phase source as an input. We investigated the result for the thresholds K = IO, 30.

5. Conclusion

We proposed an approximate analytical model for measuring the statistical Quality-of-Service (SQoS) in the high speed networks, in particular, for an ATM switch node.

We first derived upper bounds for the tail distribu- tions of the queue occupancy probability and cell delay loss problability.

As an application of the analytical model, we considered the effective service rate of the server which satisfies the required statistical QoS. Next, we derived an approximate formula for the consecutive time period during which the queue length exceeds the threshold.

From the numerical experiments, we obtained the following results: The performance of the statistical QoS obtained by the approximate model agrees very closely with the results from the simulation. This illustrates that the proposed approximate model could be used as a simple method to estimate the statistical QoS measure for the ATM switch.

Acknowledgements

The authors would like to thank the anonymous reviewers for their comments. Their remarks im- proved the accuracy and presentation of this paper.

Appendix A. The stationarity discussion of for- mula (7)

The stationarity condition for the sequence {Z,,) is that E[Z,] < 0 as n -+ x~ If we assume that there exists a stationary ergodic sequence (u,} such that lim,,, Pr{u, =Z,, V n 2 m} = 1. Let {U,} be a sequence sati,sfying the recursion U,, , = Max[ Urn + urn, 0], for n 2 1, where U, 2 0 is a finite random variable. Then, since the sequence {u,} is stationary and ergodic, there also exists, E[u] < 0, and there exists a stationary sequence {U,} with generic ele- ment given by X, such that X < m a.s. (almost

surely), and such that lim, -f li Pr(U, = U, n 2 m) = 1, for any initial value U,. Then, we finally have lim,,, Pr{ X, = U, n 2 m} = 1, for any initial value

Xl.

Appendix B. Upper bound for a primitive matrix

For a given n X n matrix F, let us assume a real constant E > 0. Then, there is a constant p = p(F, E) such that

I(F S p(sr(F) + l )” for all k = 1,2,. . . , and ij = 1,2,. . ,n.

Proof. Since the matrix F = (sr(F + E)-’ F has a spectral radius strictly less than 1, it is convergent as., and hence F” + 0 as k + x.

In particular, the elements of the sequence F’ are bounded, so there exists a finite p > 0 such that

IF,;15 P

for all k= 1,2 ,..., and i,j= 1,2 ,..., n. 0

Appendix C. Perron-Frobenius theorem

Let us summarize the Perron-Frobenius theorem as follows [4]: Let F be nonnegative and primitive as we had proved. Then, F possesses an eigenvalue r (called the Perrun-Frobenius eigenuafue) such that: 1. T is simple, real, and positive, and r> I?) for any

other eigenvalue 7. 2. There exist left and right eigenvectors 5? and 9’

corresponding to the eigenvalue T, which have strictly positive coordinates.

3. There exists a constant matrix D which satisfies

=D>O,

where D =9;sj, i, j U, and the eigenvectors are normalized so that 9” = Cfr i 5$%‘~ = 1.

Appendix D. Queue length distribution

We derive the probability matrix for the queue state in an arbitrary time slot. Let x/(i) be the

1930 H. Lee, Y. Nemoto / Computer Networks and ISDN Systems 29 (I 997) I91 9-1931

conditional probability that the queue length is i given that the source has a phase j at time slot k, which is given by X/(i) = Pr(X, = ilG, = j}, and let XL(i) be its corresponding matrix

x,(i) =(X:(i), ...3Xli”Ci))‘. (33) Let us also define the conditional probability for the cell arrival by y’(i) = Pr(Y, = i\G, = j), and F’(i) = Pr(Y, 2 ilG, = j}. Let the matrix Y(i) = diag(y’(i),y?i), . . . , y”(i))PT, where P = (Pi,j) = Pr{G, = jlGt-, = i], is the state transition matrix for the phase of the source between the (k - 1)th and kth time slot. Let the resulting matrix for the arrival process Y”(i) = diag(y’(i),F*(i), . . . ,jj”(i))PT. Then, the probability matrix x,(i) is given by

c- 1 xdi) = Y(i) C x,-,(j)

j=O

+ fj Y(i-j+c)Xk-,(j),OSi<K, j=c c-l

XkW = f(K) c Xk-t(i) i=O

f i ?(K-i+c)Xkml(i), i= K, i=c

(34)

which can be solved recursively if the initial condi- tion is given. If we reflect the initial condition in Section 2, we may have

/y,(i) = eT i = 0, OT otherwise, (35)

where e is the 1 X h4 matrix with all elements equal to one and 0 is the 1 X M matrix in which all components equal to zero.

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Hoon Lee was born in Daegu, Korea, on December 20, 1961. He received the B.E. and M.E. in electronics and com- munications from KyongPook National University, Daegu, Korea, in 1984 and 1986, respectively. In February 1986, he

joined Korea Telecom Research and De- velopment Group, where he has been engaged in the research on the teletraffic theory, network planning, traffic mod- elling and management of ISDN and B-ISDN networks. He received Ph.D. in

electrical and communication engineering from Tohoku Univer- sity, Sendai. Japan, in 1996.

He is now engaged in the research work on the stochastic modelling of a queue, queue input and output control. perfor- mance modelling for the ATM networks, QoS guarantee and congestion control in broadband wired and wireless networks. Dr. Lee is a member of IEEE and the Korea Institute of Telematics and Electronics (K.ITE).

Yashiaki Nemoto was born in Sendai city, Miyagi prefecture, Japan, on De- cember 2, 1945. He received the B-E., M.E., and Dr. Eng. degrees from To- hoku University, Sendai, Japan, in 1968, 1970, and 1973, respectively. From 1973 to 1984 he was a research associate with the Faculty of Engineering, Tohoku University. From 1984 to 199L, he was an Associate Professor with the Re- search Institute of Electrical Commum- cation in the same university.

Since 1994, he has been a Professor at Computer Center and at present he is a Professor in the Graduate School of Information Sciences, Tohoku University.

He has been engaged in the research work on microwave network, communication systems, computer networking, image processing, hand written character recognition and computer net- work management. In 1982. he received the Microwave Prize from the IEEE MTT-society. Dr. Nemeto is a member of the IEEE, the Institute of Electronics, Information and Communica- tion Engineers in Japan, and the Information Processing Society of Janan.