- Email: [email protected]
Heavy traffic analysis of integrated services systems
PERFORMANCE EVALUATION plntamational Performance
ELSEVIER
Evaluation
24 (1996) 295-302
Heavy traffic analysis of integrated services systems Gennadi Falin Department of Probability, Mechanics and Mathematics Faculty, Moscow State Unk!ersity, Moscow 119899, Russian Federation
Received 15 March 1993
Abstract
The performance analysis of integrated voice and data systems with movable boundary, ATM multiplexers, etc. is often based on Neuts’ classical theory of Markov chains with matrix-geometric invariant measure or on more general theory of structured stochastic matrices of M/G/l type. But under heavy traffic algorithmic approaches based on these theories cannot be used for computational purposes. To analyze the systems behavior under heavy traffic we suggest the use of limit theorems for scaled performance characteristics and develop a simple method to find asymptotic behavior of the mean characteristics. Keywords: Stationary
distribution;
Lyapunov function; Inequalities;
Heavy traffic; Limit theorem
1. Introduction
The performance analysis of integrated voice and data systems with movable boundary, ATM multiplexers and many other modern communication systems (see, e.g. [7,8,11]) is often based on Neuts’ classical theory [5] of Markov chains with matrix-geometric invariant measure. More complex models are analyzed with the help of Neuts’ recent generalization [6] to structured stochastic matrices of M/G/l type (see, e.g. [4,12]). It should be noted, however, that under heavy traffic Neuts [6, p. 1591, does not recommend computation of the stationary characteristics with the help of the algorithmic approaches based on his theories [5,6]. In the simplest case, heavy traffic means that the arrival rate A is close to the system capacity C. Then performance characteristics like the mean queue length, 4 = q(h), tend to infinity. But there usually exists a limit lim h,cq(A)(C - A) = Q, where the constant Q can be found in a relatively simple form. Thus under heavy traffic we can approximate the function q(A) with the help of a far simpler function Q/
Science
B.V. All rights reserved
296
G. Falin /Performance
Evaluation 24 (1996) 295-302
that the capacity C does not depend on p. Then under heavy traffic graphics of all functions 4(A, p) are almost vertical and coincide with the vertical line h = C. Obviously, such curves are useless in practice. The point is that under heavy traffic the queue length must be measured on an other scale. Expressed mathematically this means that the normalized queue (C - h)q(A, p,) should be considered rather than the ordinary queue q(h, p). Besides, the limit constant Q = Q(p) can be used for the purpose of asymptotic optimization with respect to the parameter p. Thus, the contribution of this paper consists in using heavy traffic limit theorems for asymptotic analysis of performance characteristics of integrated services systems. To avoid unnecessary complication with minor details, we restrict ourselves to a concrete model, namely, a model of an integrated services TDM system which was investigated by Yang and Mark [12]. 2. Model description Consider a wideband trunk with discrete time t = 0, 1, 2,. . . which is used to transmit two types of packetshigh priority and low priority. Each frame (t, t + 1) is divided into M slots and exactly one packet can be transmitted at a slot. Let A,(t) and A,(t) be the number of new high priority and low priority packets entering the system during the frame (t - 1, t). We assume that (1) the sequence A,(t), t 2 0, forms a Markov chain with a finite state space {O, 1,. . . , N}, one-step transition matrix P = ( pn,Jo ~ n,mI N and stationary distribution TV; (2) the sequence A,(t), I 2 0, consists of i.i.d. random variables with finite mean A and variance (T2. It should be noted that it is assumed in [12] that the high priority packets are generated by N independent sources each of which, when active, generates one packet. Activity periods have a geometric distribution with parameter p, and silence periods have a geometric distribution with parameter p,, but for the mathematical analysis only the total number A,(t) of active high priority sources is of interest. The specific structure of the variable A,(t) does not matter. The high priority packets cannot be queued. Thus if Ah(t) I M, then all A,(t) packets are transmitted at time t. But if A,(t) > M, then only M high priority packets are transmitted at time t and the rest A,(t) - it4 packets are lost and do not have an influence on further system functioning. Low priority packets can be queued in an infinite buffer. Service capacity available at time t to the low priority traffic is equal to the number of empty slots at the frame (t, t + l), i.e. is equal to s(t) = max(0, M-A,(t)). Let t(t) be the number of low priority packets in the queue at the time t. We assume that l(t) is measured just before the time when s(t) packets are chosen for transmission, but after the time when A,(t) packets joined the queue. The process t(t) is the most important process associated with the system under consideration. Obviously the following recursive formula holds: s(t + 1) = (s(t) - s(t)>‘+A,(t where (a) + = max(O, a).
+ I),
(1)
G. Falin /Performance
Eualuation 24 (I 996) 295-302
Equation (1) implies that the two-dimensional process (l(t), lattice semi-strip Z+X{O, . . . , N} as the state space.
291
A,(t)) is a Markov chain with a
3. Ergodicity Since we will investigate the system under heavy traffic we need to know the system capacity C (with respect to the low priority traffic). Using mathematical language this means that we need to know the necessary and sufficient condition for ergodicity of the chain (t(t), A,(t)). A necessary condition for ergodicity can be obtained as follows. Since the low priority packets cannot be lost, in steady state the mean traffic carried is equal to the mean traffic offered. But the mean traffic offered is A and the mean traffic carried is equal to the mean number of slots at a frame, which are occupied by the low priority packets. Thus the total number of busy slots at a frame is A + E min(M, A,(t)). This variable is obviously less than M and therefore the inequality M-1 A <
c
7Tn(M-n)
(2)
n=O
is necessary for ergodicity. As a matter of fact condition (2) is sufficient for ergodicity. This follows from the classical general theorem about ergodicity of an G/G/l queue (Eq. (1) implies that t(t) can be thought of as waiting time of a customer No. t in the G/G/l queue with interarrival intervals s(t) and service times A,(t)). Thus the capacity of the system is M-l c=
c
7Tn(M-n).
n=O
But taking into account further considerations we will give an alternative proof based on the theory of Lyapunov functions. First recall the following well-known general result (below, E is the identity matrix). Lemma. The set of linear equations (P-E)(xO
,..., ~~)~=(b,,
,..., b,)T
has a solution iff the vector (b,, . . . , b,lT satisfies the condition : b,rn=O. II=0 Now consider the following Lyapunov (5(t), A,(t)):
function
4(i, n) on the state space of the chain
4(i, n) =a, +i, where the variables a,, are a solution of the following set of linear equations: (P-E)(a,
,..., a,)T=(s,-C,s,-C
and s, = (M - n>+.
,..., s,,,-C)~
G. Falin /Performance
298
Evaluation 24 (1996) 295-302
Since f
(sn - C)rrn = f
sn7r*- c = 0
n=O
n=O
such a solution exists. Then the mean drift Xin =E(f#(t
+ l), A& + 1)) -$(5(t),
A&))
Is(t) =i, A&)
=n)
from a point (i, n) is given by the following formula: N
-a,) c Pnm(Um
Xin =
+ (i-
(M-n)+)++h
-i.
m=O
The sum x, = C~=,,~,Jum -a,) is equal to s, - C. Indeed, the vector (x,,, . . . , xN) can be expressed as (P - E)(a,, . . . , u,,,)~ and thus it is equal to (sO - C, . . . , SN- C). Therefore Xin = i
A-C,
if i2 (M-n)+,
A-C+s,-i,
ifOIiI(M-n)+-1.
(3)
Since the set {(i, n) (0 I i I (M - n)+} is finite, applying the classic Foster criteria [9], we can guarantee that inequality (2) implies the ergodicity of the chain (t(t), A,(t)). Besides, since down drifts of the chain are bounded below, it follows from [lo] that inequality (2) is necessary for ergodicity. From now on we will assume that condition (2) holds and that the process (t(t), A,(t)) is stationary. Let n,, = Pr([(t) = i, A,(t) = n> be its stationary distribution.
4. Heavy traffic limit theorem for the mean queue length In this subsection we investigate asymptotics of the mean queue length q(h) = E[(t) of the low priority packets under heavy traffic, when A + C-. Some information about behavior of the queue can be obtained with the help of Lyapunov function $(i, n) we have used to investigate the ergodicity. Namely, the well-known mean drift relation [lo] c ni,xi,
=0
(i,n)
becomes (we use Eq. (3) for the mean drifts xin) M-l M-n-l C C 17,,(M-T.Z-i)=C-A. n=O
i=O
(4)
Thus, as A -+ C- the probabilities nin for 0 I i I (A4 - n)+- 1 tend to 0. Since the chain ([(tl,Ah(t)l is irreducible, all probabilities ni, tend to 0, i.e. t(t) + +m. Although this result is obvious, we gave this proof to demonstrate the use of the mean drift relation.
G. Falin /Performance
Evaluation
24 (1996) 295-302
299
Remark 1. To apply the mean-drift relation from [lo] we must assume that Cn,,@(i, n) < co, i.e. Et(t) < 03.Due to [6,3.3], this is the case iff a2 < ~0.In fact, a direct analysis of Kolmogorov equations for the stationary distribution n,, shows that (4) holds without the assumption CT2
*(i,
where the variables a,, were introduced set of linear equations: b,)T=
(P-E)@,,...,
in Section 3, and the variables b, satisfy the following
-(fo,...,fh,)T+
2 ~,J,,(l,...J)’ n=O
and f, = -s,2 - 2(a,s,
- Aa, - Cs, + AC) + u2 + A2.
(5)
Note that due to the above lemma such a solution exists. The mean drift Yj, = qb(S(f
+ 11,
&(t
+ 1))
- 4(5(t),
4(t))
I sw
= i, 4(f)
= n>
is given by Yin= t
P,,(b,
- b,) + 2 m~OPnm[(A+(i-51)+)am-ian] E
+ [(i-sn)+12
m=O +
2A(i -s,,)+ + a2 + A2 - i2.
Using the concrete form of the variables b, the sum C~,,p,,( -f, + C,“_,~,T~. Thus -2i(C-A)
+ f fmrm, t?* =0
Yin =
s,!f- i2 + 2a,(sn -i)
b, - b,) can be transformed
to
if i2 (M-n)+,
- 2s,(C -A) + t
f,nm,
if is(M-n)+-
1
m=O
I Applying the mean drift relation we get M-l
2(C
-
+Wt)
=
? fmrm m=O
+
Remark 2. To apply the mean-drift
c n=O
M-n-l c
17,,(s,-i)(s,+i-2(C-A-a,)).
(6)
i=o
relation from [lo] we must assume that CII,,t,h(i, n) < M, i.e. E[[(t)12 < ~0. Due to [6,3.3], this is the case iff ~[A,(tll~ < 00. In fact, a direct analysis of Kolmogorov equations for the stationary distribution 17i, shows that (4) holds under the assumption a2 < w only.
G. Falin /Performance
300
Evaluation 24 (1996) 295-302
Now let A + C-. As we established with the help of (4), all probabilities LJi, for 0 5 i 5 1 tend to 0. Thus the second term in the right-hand side of (6) tends to 0 also. The limit of the first term can be calculated directly with the help of (5):
(M - n)+-
where (TV= lim, ~ c u. Thus the following main theorem holds. Theorem 1. Under heavy traffic, as A + C-, there exists lim (C-A)Et(t)=
M-l
u;+c2 2
A-C-
- + C
7rJM-n)2+
f
n=O
7rnan(C-
(M-n)+).
n=O
(7)
Corollary. Asymptotically the mean length of the low priority queue depends on the distribution of the number of new arrivals of low priority packets during a frame only through its mean A and variance u 2. The above analysis can be made more precise to get some inequalities for Et(t). Consider again as a base equation (6). The only unknown term there is the second term in the right-hand side where the unknown probabilities Lri, are involved. But we have an information about these probabilities which is given by (4). To reduce this term to (4) denote a, = min(a,, . . . , aN), a * = max(a,, . . . , aN). Then for n = 0,. . . , A4 - 1 and i = 0,...,M - n - 1 we have 1+2a,
-2(C-A)
~s,+i+2a,-2(C-A)I2M-1+2a*-2(C-A).
Taking (4) into account from this we get M-l
(C-A)[1+2a.
-2(c-A)]
s
C n=O
M-n-l
C
17,,(S,-i)(s,+i-2(C-A-a,))
i=O
r(C-A)[2M-1+2a*-2(C-A)]. Now (6) implies that the following theorem holds. Theorem 2. The mean length of the queue of low priority packets satisfies the following inequality: N u2
a, +i+ArEt(t)-
+
A2-
c rfl(si + 2a,s, n=O 2(C-A)
- 2Aa,)
+M-++A.
(8)
Note that this inequality is asymptotically exact, and as A + C- it gives Theorem 1. It should be noted also that both asymptotic formula (7) and inequality (8) involve only the stationary distribution r,, of the number of high priority packets arriving during a frame and the variables a,, 0 I n I N. Thus to apply our results we have to solve only two sets of linear equations with the same matrix, namely, (r,,, . . . , rN)(P -E) = 0 and (P - E)(aO,. . . , aNjT = (s,, - C, . . . , sN CIT. This is far simpler than the general algorithmic procedures [5,6].
G. Falin / Performance Evaluation 24 (1996) 295-302
301
Table 1 Mean low orioritv queue length
q(A)
h/C 0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
Exact Approx. Lower bound Upper bound
8.3 8.9 7.5 10.5
9.3 9.9 8.5 11.5
10.5 11.1 9.7 12.7
12.1 12.7 11.3 14.3
14.2 14.8 13.4 16.4
17.2 17.8 16.4 19.4
21.6 22.2 20.8 23.8
29.0 29.6 28.2 31.2
43.8 44.4 43.0 46.0
88.2 88.8 87.4 90.4
5. A numerical
example
To illustrate the quality of our approximation and inequalities for the mean number of low priority packets in the queue we consider the system with M = 2 slots in the frame. High priority packets are generated by N = 3 independent sources each of which when active generates one packet. Active periods have a geometric distribution with parameter p,, = 0.5 and silence periods have a geometric distribution with parameter p, = 0.5. In this case the sequence A,(t), t=o, l)...) consists of independent and identically distributed random variables, and ?ro = l/8, rrr = 3/8, Vz = 3/8, 7rs = l/8. Obviously, s0 = 2, s1 = 1, s2 = 0, s3 = 0 and it is easy to show that a, = 0, a, = 1, a2 = 2, a3 = 2 (thus a * = 0, a * = 2). Low priority packets arrive in accordance with a Poisson flow with the rate A packets/frame (thus a2 = A). The system capacity is C = 27r,, + r1 = 5/S Now (7) implies, that under heavy traffic q(A) = 71/(80 128Al. In Table 1 we give the exact (obtained with the help of simulation) and approximate (obtained with the help of (7)) values of the mean queue length. We also give the upper and lower bounds for q(A) obtained with the help of (8). In this example, A/C has been varied from 0.90 up to 0.99. The relative error is about 6% if A/C = 0.90 and about 0.25% if A/C = 0.99. 6. Conclusion In the present paper we suggested the use of limit theorems for analysis of performance characteristics of integrated services systems under heavy traffic. We developed a simple method based on Lyapunov functions to find the asymptotic behavior of mean characteristics. We established also an inequality for performance characteristics. Since the classic algorithmic approaches are not recommended for use under heavy traffic, these results seem to be an attractive alternative for obtaining numerical results. It should be noted that we investigated only the mean characteristic of the system. It is interesting to investigate asymptotic behavior of the distribution of the low priority queue. This can be done with the help of an approach developed in [l-3] for continuous time queueing systems with randomly varying service rate. Acknowledgement
The research described in this publication was made possible in part by Grant No. NCGOOO from the International Science Foundation.
302
G. FaIin/Pe&rmance
Evaluation 24 (1996) 295-302
References [l] G. Falin, On a single server system with randomly varying service rate, Soviet J. Comput. Systems Sci. 3 (1989) 26-28. [2] G. Falin, Analysis of a buffered floating-threshold hybrid switching system, Probl. Inform. Transmission 24 (4) (1988) 318-323. [3] G. Falin, Random walks on semi-strip and hybrid switching systems with movable boundary, Proc. Int. Seminar on Teletraffic Theory and Computer Modelling, Sofia, Bulgaria, 1988, pp. 29-40 (in Russian).
[4] C.K. Jeong and C.K. Un, Performance analysis of a voice/data multiplexer based on Markov renewal process modelling, Proc. IEEE 76 (10) (1988) 1390-1393. [5] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach John Hopkins University Press, Baltimore, Md. (1981). [6] M.F. Neuts, Structured Stochastic Matrices of M/G/l Type and Their Applications, Marcel Dekker, New York (1989). [7] B. Ngo and H. Lee, Queueing analysis of traffic access control strategies with preemptive and nonpreemptive disciplines in wideband integrated networks, IEEE J. Select. Areas Comm. 9 (7) (1991) 1093-1109. [8] Z. Niu and H. Akimaru, Analysis of statistical multiplexer with selective cell discarding control in ATM systems, IEZCE Trans. 74 (12) (1991) 4069-4079. [9] A.G. Pakes, Some conditions for ergodicity and recurrence
of Markov chains, Op. Res. 17 (1969) 1058-1061. [lo] L.I. Sennott, P.A. Humblet and R.L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Op. Res. 31 (4) (1983) 783-789. [ll] Y. De Serres and L.G. Mason, A multiserver queue with narrow-and wide-band customers and wide-band restricted access, IEEE Trans. Comm. 36 (1988) 675-684. 1121 W. Yang and J.W. Mark, Queueing analysis of an integrated services TDM system using a matrix-analytic method. IEEE J. Select. Areas Comm. 9 (1) (1991) 88-93. G. Falin graduated from Moscow State University in 1973 with honours. He received his Ph.D. and Doctor of Sciences degrees in probability and mathematical statistics from Moscow State University in 1979 and 1989, respectively. In 1979 he joined the Department of Probability of the Mechanics and Mathematics Faculty at the Moscow State University, where currently holds the position of The Leading Scientist. Since joining the Department of Probability, Dr. Falin has been involved in research projects with Soviet telecommunication companies and research centers as coordinator and (since 1990) as Scientific Manager of the Teletraffic Research Group. Dr. Falin has published over 70 articles on retrial queues, circuit switching networks, stochastic orderings, slotted ALOHA and other queueing problems. He is a member of the editorial board of Communications in Statistics, Stochastic Models and a member of the Moscow Mathematical Society.
ELSEVIER
Evaluation
24 (1996) 295-302
Heavy traffic analysis of integrated services systems Gennadi Falin Department of Probability, Mechanics and Mathematics Faculty, Moscow State Unk!ersity, Moscow 119899, Russian Federation
Received 15 March 1993
Abstract
The performance analysis of integrated voice and data systems with movable boundary, ATM multiplexers, etc. is often based on Neuts’ classical theory of Markov chains with matrix-geometric invariant measure or on more general theory of structured stochastic matrices of M/G/l type. But under heavy traffic algorithmic approaches based on these theories cannot be used for computational purposes. To analyze the systems behavior under heavy traffic we suggest the use of limit theorems for scaled performance characteristics and develop a simple method to find asymptotic behavior of the mean characteristics. Keywords: Stationary
distribution;
Lyapunov function; Inequalities;
Heavy traffic; Limit theorem
1. Introduction
The performance analysis of integrated voice and data systems with movable boundary, ATM multiplexers and many other modern communication systems (see, e.g. [7,8,11]) is often based on Neuts’ classical theory [5] of Markov chains with matrix-geometric invariant measure. More complex models are analyzed with the help of Neuts’ recent generalization [6] to structured stochastic matrices of M/G/l type (see, e.g. [4,12]). It should be noted, however, that under heavy traffic Neuts [6, p. 1591, does not recommend computation of the stationary characteristics with the help of the algorithmic approaches based on his theories [5,6]. In the simplest case, heavy traffic means that the arrival rate A is close to the system capacity C. Then performance characteristics like the mean queue length, 4 = q(h), tend to infinity. But there usually exists a limit lim h,cq(A)(C - A) = Q, where the constant Q can be found in a relatively simple form. Thus under heavy traffic we can approximate the function q(A) with the help of a far simpler function Q/
Science
B.V. All rights reserved
296
G. Falin /Performance
Evaluation 24 (1996) 295-302
that the capacity C does not depend on p. Then under heavy traffic graphics of all functions 4(A, p) are almost vertical and coincide with the vertical line h = C. Obviously, such curves are useless in practice. The point is that under heavy traffic the queue length must be measured on an other scale. Expressed mathematically this means that the normalized queue (C - h)q(A, p,) should be considered rather than the ordinary queue q(h, p). Besides, the limit constant Q = Q(p) can be used for the purpose of asymptotic optimization with respect to the parameter p. Thus, the contribution of this paper consists in using heavy traffic limit theorems for asymptotic analysis of performance characteristics of integrated services systems. To avoid unnecessary complication with minor details, we restrict ourselves to a concrete model, namely, a model of an integrated services TDM system which was investigated by Yang and Mark [12]. 2. Model description Consider a wideband trunk with discrete time t = 0, 1, 2,. . . which is used to transmit two types of packetshigh priority and low priority. Each frame (t, t + 1) is divided into M slots and exactly one packet can be transmitted at a slot. Let A,(t) and A,(t) be the number of new high priority and low priority packets entering the system during the frame (t - 1, t). We assume that (1) the sequence A,(t), t 2 0, forms a Markov chain with a finite state space {O, 1,. . . , N}, one-step transition matrix P = ( pn,Jo ~ n,mI N and stationary distribution TV; (2) the sequence A,(t), I 2 0, consists of i.i.d. random variables with finite mean A and variance (T2. It should be noted that it is assumed in [12] that the high priority packets are generated by N independent sources each of which, when active, generates one packet. Activity periods have a geometric distribution with parameter p, and silence periods have a geometric distribution with parameter p,, but for the mathematical analysis only the total number A,(t) of active high priority sources is of interest. The specific structure of the variable A,(t) does not matter. The high priority packets cannot be queued. Thus if Ah(t) I M, then all A,(t) packets are transmitted at time t. But if A,(t) > M, then only M high priority packets are transmitted at time t and the rest A,(t) - it4 packets are lost and do not have an influence on further system functioning. Low priority packets can be queued in an infinite buffer. Service capacity available at time t to the low priority traffic is equal to the number of empty slots at the frame (t, t + l), i.e. is equal to s(t) = max(0, M-A,(t)). Let t(t) be the number of low priority packets in the queue at the time t. We assume that l(t) is measured just before the time when s(t) packets are chosen for transmission, but after the time when A,(t) packets joined the queue. The process t(t) is the most important process associated with the system under consideration. Obviously the following recursive formula holds: s(t + 1) = (s(t) - s(t)>‘+A,(t where (a) + = max(O, a).
+ I),
(1)
G. Falin /Performance
Eualuation 24 (I 996) 295-302
Equation (1) implies that the two-dimensional process (l(t), lattice semi-strip Z+X{O, . . . , N} as the state space.
291
A,(t)) is a Markov chain with a
3. Ergodicity Since we will investigate the system under heavy traffic we need to know the system capacity C (with respect to the low priority traffic). Using mathematical language this means that we need to know the necessary and sufficient condition for ergodicity of the chain (t(t), A,(t)). A necessary condition for ergodicity can be obtained as follows. Since the low priority packets cannot be lost, in steady state the mean traffic carried is equal to the mean traffic offered. But the mean traffic offered is A and the mean traffic carried is equal to the mean number of slots at a frame, which are occupied by the low priority packets. Thus the total number of busy slots at a frame is A + E min(M, A,(t)). This variable is obviously less than M and therefore the inequality M-1 A <
c
7Tn(M-n)
(2)
n=O
is necessary for ergodicity. As a matter of fact condition (2) is sufficient for ergodicity. This follows from the classical general theorem about ergodicity of an G/G/l queue (Eq. (1) implies that t(t) can be thought of as waiting time of a customer No. t in the G/G/l queue with interarrival intervals s(t) and service times A,(t)). Thus the capacity of the system is M-l c=
c
7Tn(M-n).
n=O
But taking into account further considerations we will give an alternative proof based on the theory of Lyapunov functions. First recall the following well-known general result (below, E is the identity matrix). Lemma. The set of linear equations (P-E)(xO
,..., ~~)~=(b,,
,..., b,)T
has a solution iff the vector (b,, . . . , b,lT satisfies the condition : b,rn=O. II=0 Now consider the following Lyapunov (5(t), A,(t)):
function
4(i, n) on the state space of the chain
4(i, n) =a, +i, where the variables a,, are a solution of the following set of linear equations: (P-E)(a,
,..., a,)T=(s,-C,s,-C
and s, = (M - n>+.
,..., s,,,-C)~
G. Falin /Performance
298
Evaluation 24 (1996) 295-302
Since f
(sn - C)rrn = f
sn7r*- c = 0
n=O
n=O
such a solution exists. Then the mean drift Xin =E(f#(t
+ l), A& + 1)) -$(5(t),
A&))
Is(t) =i, A&)
=n)
from a point (i, n) is given by the following formula: N
-a,) c Pnm(Um
Xin =
+ (i-
(M-n)+)++h
-i.
m=O
The sum x, = C~=,,~,Jum -a,) is equal to s, - C. Indeed, the vector (x,,, . . . , xN) can be expressed as (P - E)(a,, . . . , u,,,)~ and thus it is equal to (sO - C, . . . , SN- C). Therefore Xin = i
A-C,
if i2 (M-n)+,
A-C+s,-i,
ifOIiI(M-n)+-1.
(3)
Since the set {(i, n) (0 I i I (M - n)+} is finite, applying the classic Foster criteria [9], we can guarantee that inequality (2) implies the ergodicity of the chain (t(t), A,(t)). Besides, since down drifts of the chain are bounded below, it follows from [lo] that inequality (2) is necessary for ergodicity. From now on we will assume that condition (2) holds and that the process (t(t), A,(t)) is stationary. Let n,, = Pr([(t) = i, A,(t) = n> be its stationary distribution.
4. Heavy traffic limit theorem for the mean queue length In this subsection we investigate asymptotics of the mean queue length q(h) = E[(t) of the low priority packets under heavy traffic, when A + C-. Some information about behavior of the queue can be obtained with the help of Lyapunov function $(i, n) we have used to investigate the ergodicity. Namely, the well-known mean drift relation [lo] c ni,xi,
=0
(i,n)
becomes (we use Eq. (3) for the mean drifts xin) M-l M-n-l C C 17,,(M-T.Z-i)=C-A. n=O
i=O
(4)
Thus, as A -+ C- the probabilities nin for 0 I i I (A4 - n)+- 1 tend to 0. Since the chain ([(tl,Ah(t)l is irreducible, all probabilities ni, tend to 0, i.e. t(t) + +m. Although this result is obvious, we gave this proof to demonstrate the use of the mean drift relation.
G. Falin /Performance
Evaluation
24 (1996) 295-302
299
Remark 1. To apply the mean-drift relation from [lo] we must assume that Cn,,@(i, n) < co, i.e. Et(t) < 03.Due to [6,3.3], this is the case iff a2 < ~0.In fact, a direct analysis of Kolmogorov equations for the stationary distribution n,, shows that (4) holds without the assumption CT2
*(i,
where the variables a,, were introduced set of linear equations: b,)T=
(P-E)@,,...,
in Section 3, and the variables b, satisfy the following
-(fo,...,fh,)T+
2 ~,J,,(l,...J)’ n=O
and f, = -s,2 - 2(a,s,
- Aa, - Cs, + AC) + u2 + A2.
(5)
Note that due to the above lemma such a solution exists. The mean drift Yj, = qb(S(f
+ 11,
&(t
+ 1))
- 4(5(t),
4(t))
I sw
= i, 4(f)
= n>
is given by Yin= t
P,,(b,
- b,) + 2 m~OPnm[(A+(i-51)+)am-ian] E
+ [(i-sn)+12
m=O +
2A(i -s,,)+ + a2 + A2 - i2.
Using the concrete form of the variables b, the sum C~,,p,,( -f, + C,“_,~,T~. Thus -2i(C-A)
+ f fmrm, t?* =0
Yin =
s,!f- i2 + 2a,(sn -i)
b, - b,) can be transformed
to
if i2 (M-n)+,
- 2s,(C -A) + t
f,nm,
if is(M-n)+-
1
m=O
I Applying the mean drift relation we get M-l
2(C
-
+Wt)
=
? fmrm m=O
+
Remark 2. To apply the mean-drift
c n=O
M-n-l c
17,,(s,-i)(s,+i-2(C-A-a,)).
(6)
i=o
relation from [lo] we must assume that CII,,t,h(i, n) < M, i.e. E[[(t)12 < ~0. Due to [6,3.3], this is the case iff ~[A,(tll~ < 00. In fact, a direct analysis of Kolmogorov equations for the stationary distribution 17i, shows that (4) holds under the assumption a2 < w only.
G. Falin /Performance
300
Evaluation 24 (1996) 295-302
Now let A + C-. As we established with the help of (4), all probabilities LJi, for 0 5 i 5 1 tend to 0. Thus the second term in the right-hand side of (6) tends to 0 also. The limit of the first term can be calculated directly with the help of (5):
(M - n)+-
where (TV= lim, ~ c u. Thus the following main theorem holds. Theorem 1. Under heavy traffic, as A + C-, there exists lim (C-A)Et(t)=
M-l
u;+c2 2
A-C-
- + C
7rJM-n)2+
f
n=O
7rnan(C-
(M-n)+).
n=O
(7)
Corollary. Asymptotically the mean length of the low priority queue depends on the distribution of the number of new arrivals of low priority packets during a frame only through its mean A and variance u 2. The above analysis can be made more precise to get some inequalities for Et(t). Consider again as a base equation (6). The only unknown term there is the second term in the right-hand side where the unknown probabilities Lri, are involved. But we have an information about these probabilities which is given by (4). To reduce this term to (4) denote a, = min(a,, . . . , aN), a * = max(a,, . . . , aN). Then for n = 0,. . . , A4 - 1 and i = 0,...,M - n - 1 we have 1+2a,
-2(C-A)
~s,+i+2a,-2(C-A)I2M-1+2a*-2(C-A).
Taking (4) into account from this we get M-l
(C-A)[1+2a.
-2(c-A)]
s
C n=O
M-n-l
C
17,,(S,-i)(s,+i-2(C-A-a,))
i=O
r(C-A)[2M-1+2a*-2(C-A)]. Now (6) implies that the following theorem holds. Theorem 2. The mean length of the queue of low priority packets satisfies the following inequality: N u2
a, +i+ArEt(t)-
+
A2-
c rfl(si + 2a,s, n=O 2(C-A)
- 2Aa,)
+M-++A.
(8)
Note that this inequality is asymptotically exact, and as A + C- it gives Theorem 1. It should be noted also that both asymptotic formula (7) and inequality (8) involve only the stationary distribution r,, of the number of high priority packets arriving during a frame and the variables a,, 0 I n I N. Thus to apply our results we have to solve only two sets of linear equations with the same matrix, namely, (r,,, . . . , rN)(P -E) = 0 and (P - E)(aO,. . . , aNjT = (s,, - C, . . . , sN CIT. This is far simpler than the general algorithmic procedures [5,6].
G. Falin / Performance Evaluation 24 (1996) 295-302
301
Table 1 Mean low orioritv queue length
q(A)
h/C 0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
Exact Approx. Lower bound Upper bound
8.3 8.9 7.5 10.5
9.3 9.9 8.5 11.5
10.5 11.1 9.7 12.7
12.1 12.7 11.3 14.3
14.2 14.8 13.4 16.4
17.2 17.8 16.4 19.4
21.6 22.2 20.8 23.8
29.0 29.6 28.2 31.2
43.8 44.4 43.0 46.0
88.2 88.8 87.4 90.4
5. A numerical
example
To illustrate the quality of our approximation and inequalities for the mean number of low priority packets in the queue we consider the system with M = 2 slots in the frame. High priority packets are generated by N = 3 independent sources each of which when active generates one packet. Active periods have a geometric distribution with parameter p,, = 0.5 and silence periods have a geometric distribution with parameter p, = 0.5. In this case the sequence A,(t), t=o, l)...) consists of independent and identically distributed random variables, and ?ro = l/8, rrr = 3/8, Vz = 3/8, 7rs = l/8. Obviously, s0 = 2, s1 = 1, s2 = 0, s3 = 0 and it is easy to show that a, = 0, a, = 1, a2 = 2, a3 = 2 (thus a * = 0, a * = 2). Low priority packets arrive in accordance with a Poisson flow with the rate A packets/frame (thus a2 = A). The system capacity is C = 27r,, + r1 = 5/S Now (7) implies, that under heavy traffic q(A) = 71/(80 128Al. In Table 1 we give the exact (obtained with the help of simulation) and approximate (obtained with the help of (7)) values of the mean queue length. We also give the upper and lower bounds for q(A) obtained with the help of (8). In this example, A/C has been varied from 0.90 up to 0.99. The relative error is about 6% if A/C = 0.90 and about 0.25% if A/C = 0.99. 6. Conclusion In the present paper we suggested the use of limit theorems for analysis of performance characteristics of integrated services systems under heavy traffic. We developed a simple method based on Lyapunov functions to find the asymptotic behavior of mean characteristics. We established also an inequality for performance characteristics. Since the classic algorithmic approaches are not recommended for use under heavy traffic, these results seem to be an attractive alternative for obtaining numerical results. It should be noted that we investigated only the mean characteristic of the system. It is interesting to investigate asymptotic behavior of the distribution of the low priority queue. This can be done with the help of an approach developed in [l-3] for continuous time queueing systems with randomly varying service rate. Acknowledgement
The research described in this publication was made possible in part by Grant No. NCGOOO from the International Science Foundation.
302
G. FaIin/Pe&rmance
Evaluation 24 (1996) 295-302
References [l] G. Falin, On a single server system with randomly varying service rate, Soviet J. Comput. Systems Sci. 3 (1989) 26-28. [2] G. Falin, Analysis of a buffered floating-threshold hybrid switching system, Probl. Inform. Transmission 24 (4) (1988) 318-323. [3] G. Falin, Random walks on semi-strip and hybrid switching systems with movable boundary, Proc. Int. Seminar on Teletraffic Theory and Computer Modelling, Sofia, Bulgaria, 1988, pp. 29-40 (in Russian).
[4] C.K. Jeong and C.K. Un, Performance analysis of a voice/data multiplexer based on Markov renewal process modelling, Proc. IEEE 76 (10) (1988) 1390-1393. [5] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach John Hopkins University Press, Baltimore, Md. (1981). [6] M.F. Neuts, Structured Stochastic Matrices of M/G/l Type and Their Applications, Marcel Dekker, New York (1989). [7] B. Ngo and H. Lee, Queueing analysis of traffic access control strategies with preemptive and nonpreemptive disciplines in wideband integrated networks, IEEE J. Select. Areas Comm. 9 (7) (1991) 1093-1109. [8] Z. Niu and H. Akimaru, Analysis of statistical multiplexer with selective cell discarding control in ATM systems, IEZCE Trans. 74 (12) (1991) 4069-4079. [9] A.G. Pakes, Some conditions for ergodicity and recurrence
of Markov chains, Op. Res. 17 (1969) 1058-1061. [lo] L.I. Sennott, P.A. Humblet and R.L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Op. Res. 31 (4) (1983) 783-789. [ll] Y. De Serres and L.G. Mason, A multiserver queue with narrow-and wide-band customers and wide-band restricted access, IEEE Trans. Comm. 36 (1988) 675-684. 1121 W. Yang and J.W. Mark, Queueing analysis of an integrated services TDM system using a matrix-analytic method. IEEE J. Select. Areas Comm. 9 (1) (1991) 88-93. G. Falin graduated from Moscow State University in 1973 with honours. He received his Ph.D. and Doctor of Sciences degrees in probability and mathematical statistics from Moscow State University in 1979 and 1989, respectively. In 1979 he joined the Department of Probability of the Mechanics and Mathematics Faculty at the Moscow State University, where currently holds the position of The Leading Scientist. Since joining the Department of Probability, Dr. Falin has been involved in research projects with Soviet telecommunication companies and research centers as coordinator and (since 1990) as Scientific Manager of the Teletraffic Research Group. Dr. Falin has published over 70 articles on retrial queues, circuit switching networks, stochastic orderings, slotted ALOHA and other queueing problems. He is a member of the editorial board of Communications in Statistics, Stochastic Models and a member of the Moscow Mathematical Society.