( , )-Characterization Based Connection Control for ... - CiteSeerX

( ,

)-Characterization Based Connection Control

for Guaranteed Services in High Speed Networks

Song Chong



San-qi Li

Department of Electrical and Computer Engineering The University of Texas at Austin Austin, Texas 78712 January 26, 1995

Abstract

In this paper we present a method to establish real-time connections with guaranteed qualityof-services, based on per-session (; )-characterization. Under two distinctive service disciplines, rate proportional processor sharing and xed rate processor sharing, we derive tighter probabilistic bounds on per-session end-to-end average cell loss rate, which is caused by either buer over ow in the route or excessive delay at the destination. One remarkable feature of the bounding solutions is that they are solely determined by the probabilistic (; )-characterization of each session itself, independent of the network environment and other connections. To improve network resource utilization, our method is extended to allow statistical sharing of buer resources at each node. The admission control scheme presented in this paper has a great exibility in connection management since both bandwidth and buer resources can be adaptively allocated among incoming and existing sessions according to present network resource availability. The (; )-characterization strongly depends on trac characteristics. Our study of real multimedia trac streams reveals the interrelationship among (; )-characterization, trac statistics and QOS constraint and also provides many engineering aspects of (; )-characterization for connection management.

This work was supported by National Science Foundation under grant NCR9015757 and by Texas Advanced Research Program under grant TARP-33. 

1 Introduction One of the most challenging issues in supporting multimedia communications on a high speed network is to provide quality-of-service (QOS) guarantees to sessions [1, 2]. The QOS requirements of dierent sessions can vary substantially depending on applications. The basic function of admission control is to determine whether an arriving session can be accepted at its requested QOS without violating QOS guarantees of on-going sessions. In ATM networks, the QOS is mainly measured by end-to-end cell delay and loss performance. In this paper, each session connection is de ned by (; ) where  and  respectively represent buer space and transmission bandwidth allocated for the session at each switching node in the route. Obviously, the determination of (; ) strongly depends on trac characteristics and QOS requirement. In allocating network resources, there exists a tradeo between  and . For instance, reduction of buer space can be achieved by increase of transmission bandwidth, and vice versa, as long as QOS is not violated. The (; )-characterization of a session can be de ned either deterministically or probabilistically. In the deterministic characterization, the trac of each session is viewed as a deterministic rate function x(t) of nite duration. Then, the (; )-characterization is de ned by

 = 0max [ st

Zt

s

x( )d ? (t ? s) ]; 8  2 [a; p];

(1)

where a and p are the average and peak rates of each session. Such a deterministic trac description was rst introduced by Cruz in [3, 4] and recently used by others [5]-[8]. Particularly in [7, 8], Low employed (1) to provide a tradeo between transmission bandwidth and buer space in a problem of network resource allocation. Similarly, one can introduce a probabilistic (; )-characterization for stochastic trac which is modeled by a stationary random process. Applying this random input process to a single-server work-conserving (WC) queueing system with transmission bandwidth  and in nite buer capacity, one can de ne the (; )-characterization by the following queue length steady-state probability function: Pr[ q   j  ]; 8  2 [a; p]:

(2)

In [9], Yaron and Sidi used an exponential bound of (2) for such a probabilistic trac description. One important feature of (; )-characterization is that it is determined by queueing analysis of a WC system for a given deterministic rate function or stochastic model of each session. Consider a wide class of multimedia trac to be supported on a high speed network. Our study in this paper reveals signi cant dierences among the (; )-characterizations of diverse multimedia trac. The computation of deterministic (; )-characterization is straightforward from (1), once the rate function x(t) of each session is provided. The computation of probabilistic (; )-characterization involves stochastic trac modeling and queueing analysis. In stochastic trac modeling, there are two important statistical functions: rate distribution f (x) (histogram) and power spectrum P (! ) (equivalently, autocorrelation). The former describes trac rst-order statistics and the latter captures trac second-order statistics. The queueing performance in (2) is mainly determined by f (x) and P (! ), whereas the in uence of trac higher-order statistics is negligible [14, 15]. Recently in [15], a programming approach is developed for construction of a special class of Markov modulated Poisson process (MMPP), called circulant modulated Poisson process (CMPP), to match 1

with a wide range of P (! ) and f (x). The same approach is applied here in trac modeling to re ect various statistical properties of real multimedia trac (e.g., voice, MPEG/JPEG video and LANto-LAN data). Based on this Markovian trac modeling, the numerical solution of (2) is readily obtained by using advanced queueing techniques (e.g., [16, 17]). Once network resources are allocated to each session according to its (; )-characterization, one can bound per-session end-to-end queueing/loss performance. For sessions with deterministic (; )-characterization, the QOS is measured by the deterministic guarantee of zero cell loss and bounded end-to-end delay. For sessions with probabilistic (; )-characterization, the QOS is measured by the probabilistic guarantee of overall end-to-end cell loss, which is caused by either buer over ow at intermediate nodes or excessive delay at the destination. Obviously, the tightness of per-session performance bounds is strongly dependent on the service discipline implemented at each node among dierent sessions. Methodologies have been proposed to compute per-session performance bounds under FIFO service discipline for sessions with either deterministic or probabilistic (; )-characterization [4, 9]. However, the bounds under FIFO discipline are found to be rather loose [1]. In this paper, we consider two special service disciplines which make it possible to signi cantly tighten the end-to-end performance bounds: one is called rate proportional processor sharing (RPPS) [5, 6, 10] and the other is called xed rate processor sharing (FRPS) [7, 8]. The only dierence between RPPS and FRPS is that the former allows statistical multiplexing of dierent sessions whereas the latter dedicates a xed transmission bandwidth to each session. The bound analyses under both RPPS and FRPS in [6, 7] were based on deterministic (; )-characterization. In [6], the deterministic (; )-characterization was used to describe a leaky-bucket constrained session, whereas in [7] it was used to directly describe a session as in this paper. We generalize the deterministic bounding approach in [6, 7] to a probabilistic bounding approach, based on probabilistic (; )-characterization. A simple admission control scheme can be implemented based on the probabilistic bound analysis. Since the statistical functions f (x) and P (! ) can be collected from certain representative trac streams, the (; )-characterization of each possible type of arriving sessions can be computed and stored in advance by the network trac manager. When a new session arrives, the admission control rst examines its (; )-characterization subject to the session QOS constraints and then identi es a so-called admissible set of (; ), denoted by Z . The concept of admissible set is to introduce a great exibility to resource allocation between buer space and transmission bandwidth at the connection setup stage. According to present network-wide resource availability, the admission control selects a proper pair (; ) 2 Z and reserves the corresponding resources. The QOS of the session is then guaranteed once the connection is set up. The arriving session is blocked if none of the (; )'s in Z can be ensured by the network. Another advantage of probabilistic (; )-characterization is to take into account buer sharing eect in allocating network resources. Instead of allocating a segregate buer to each connection at a link, all the connections at that link can statistically share a single large buer. By the law of large numbers, the aggregate buer space requirement will be substantially reduced while the probabilistic QOS requirement of each individual connection can still be ensured. The paper is organized as follows. In Section 2, we de ne deterministic/probabilistic ( , )characterization and derive bounds on per-session end-to-end loss performance under RPPS/FRPS service discipline. The impact of diverse multimedia trac on ( , )-characterization is examined 2

ψ i(1) session i source

φ (1) i

ψ i(2)

φ (2) i

. . .

ψ i(N i)

i) φ (N i

session i destination

(a)

Ki

µi

session i

(b)

Figure 1: (a) A RPPS/FRPS connection of session i de ned by f ((i k) ; i(k)) j k = 1; 2;


; Ni g (b) a work-conserving system transmitting session i with transmission bandwidth i and buer capacity Ki . in Section 3. The work is then extended in Section 4 to admission control design with improvement of network resource utilization by buer sharing and trac aggregation. The paper is concluded in Section 5.

2 Per-Session End-to-End Performance Consider a virtual-circuit packet-switched network where routing is performed on per-session basis. Each switching node in the network is assumed to be an output-buered switch [13]. Both RPPS and FRPS service disciplines are considered. A connection of session i is de ned by a set of connection parameters, denoted by f (mi ; im) j m 2 Ri g where Ri represents a set of links in the route, and mi and im respectively denote transmission bandwidth and buer space to be reserved at link m. Interchangeably, we also use f ((i k) ; i(k)) j k = 1; 2;


; Ni g for the same set of connection parameters where the superscript (k) represents the k-th link in the route and Ni is the number of hops. (see Fig. 1a). The total transmission bandwidth and buer capacity at link m are denoted by C m and B m , respectively. In addition to the reservation-based connections, we assume that the network also supports the so-called best-eort trac which are either connectionless or connection-oriented without resource reservation requirement. A separate buer space is assumed at each link to temporarily store the aggregate best-eort trac. Let gim(t) denote departure rate of session i at link m at time t. Under the RPPS discipline, if connection i has backlog at link m, m gim(t) = P im m C m; j 2Q j

(3)

otherwise, gim (t) = 0. Qm denotes the set of all connections \currently" having backlog at link m. Note that the RPPS is work-conserving among all connections. The background best-eort trac streams are only transmitted when all the connections have zero backlog. In contrast, under the FRPS discipline, if connection i has backlog at time t,

gim(t) = mi 3

(4)

σ

i

Pr [qi > σ i | ρi ]

σ = (D i - A i) ρi i

σ

σ = (D i - A i) ρi i

i

ρi

ρi

(a)

(b)

Figure 2: (a) Deterministic (; )-characterization (b) probabilistic (; )-characterization. regardless of the other connections. Otherwise, gim (t) = 0 and the unused bandwidth mi is taken by best-eort trac instead of other connections. First, we review the deterministic per-session performance bounds in [6, 7]. For the RPPS discipline in [6], the (; ) function (1) is used to describe a leaky-bucket constrained session, whereas in our case it is used to directly characterize a session on which no access control is imposed. Let xi (t) be the deterministic rate function of session i. When the session is transmitted through a WC system with transmission bandwidth i = i and buer capacity Ki = 1 (see Fig. 1b), the queue backlog at time t is described by

qi(t) = smax [ 2[0; t]

Zt

s

xi ( )d ? i(t ? s) ]:

(5)

Similar to (1), the buer space requirement i at each given i is equal to the maximum backlog which is expressed by

i = max [ qi (t) ] = 0max [ t st

Zt

s

xi( )d ? i(t ? s) ]:

(6)

Fig. 2a shows a typical deterministic (; )-characterization of a session i. Clearly, i is a decreasing function of i. Let q~iworst and d~worst respectively denote the worst-case end-to-end backlog and queueing delay i of session i. We use the accent ~ in the notation to represent an end-to-end connection measurement. De ne I m to be the set of connections on link mP. If a RPPS connection of session i is set up with (mi ; im ) = (i ; i ), m 2 Ri, subject to mi + j 2I m mj < C m , then [6]

d~worst  i : i

q~iworst  i ;

i

(7)

(1) (k) (k) If a FRPS connection of session i is set up with ((1) i ; i ) = (i; i ), (i ; i ) = (i; 0), k 6= 1, P subject to mi + j 2I m mj < C m , m 2 Ri , then [7]

d~worst = i : i

q~iworst = i ;

i

(8)

For the FRPS connection, no backlog ever occurs in the route except at the rst node since the bandwidth assigned at each node for the connection is xed and identical. Furthermore, both 4

RPPS and FRPS connections have zero cell loss because the buers never over ow. The bounds (7) and (8) are derived by uid ow queueing analysis. The real implementation of RPPS/FRPS connection requires an additional one-cell buer space at each link [5, 10]. In general, a user end-to-end delay consists of queueing delay, propagation delay, nodal processing delay, media compression/decompression delay and so on. Let Di be the user end-to-end delay constraint of session i, and assume that the overall non-queueing delay is bounded by Ai . For deterministic QOS guarantee, i.e. d~worst  Di ? Ai , we must assign (i; i) subject to i

i  D ? A ; i i i which leads to an admissible set for session i:

(9)

Zi = f (i ; i) j i  (Di ? Ai )i g;

(10)

as illustrated by the thickened curve in Fig. 2a. The network manager is responsible to choose a proper (i ; i) 2 Zi according to present network-wide resource availability. Let us now consider a stochastic session i whose arrival rate xi (t) is represented by a stationary random process. Applying xi (t) to a WC system with i = i and Ki , we de ne the probabilistic (; )-characterization of session i by the following queue length steady-state probability function: Pr[ qi  i j i = i ; Ki ];

(11)

which decreases with respect to i and i , as shown in Fig. 2b of a typical probabilistic (; )characterization. It also varies by Ki as described by the following proposition: Proposition. For all i; i, Pr[ qi = i j i = i; Ki = i ]  Pr[ qi  i j i = i; i  Ki < 1 ]  Pr[ qi  i j i = i; Ki = 1 ]: The proof is provided in Appendix. Unlike the deterministic session whose QOS is measured by a hard delay bound with zero cell loss, the QOS of a stochastic session is measured by a probabilistic loss bound. Cell loss can occur either in the route by buer over ow or at destination by excessive delay (i.e., end-to-end delay greater than Di ? Ai ). Denote the end-to-end cell loss probability of connection i in the route by i i . Then, the overall end-to-end cell loss probability P~ i is P~blocking and at the destination by P~delay loss i = P~ i i de ned by P~loss blocking + P~delay . Let Pi be the user-speci ed QOS requirement of session i on the overall end-to-end cell loss probability. Then, the QOS of session i is guaranteed by i = P~ i i P~loss blocking + P~delay  Pi :

The following theorem states the end-to-end loss performance bounds for RPPS connections.

Theorem 1.

P

If RPPS connection of session i is set up by (mi ; im ) = (i; i ) and mi + j 2I m mj < C m , m 2 Ri , then i  Pr[ q  minf ; (D ? A ) g j  =  ; K = 1 ]; P~loss (12) i i i i i i i i 5

and

i P~blocking  Pr[ qi  i j i = i; Ki = 1 ] i P~delay

(

qi  (Di ? Ai)i j i = i ; Ki = 1 ] if (Di ? Ai )i  Ni i  Pr[ 0 if (Di ? Ai )i > Ni i .

(13) (14)

The proof is provided in Appendix. The admissible set Zi for RPPS session i is de ned by i  P g; Zi = f (i ; i ) j P~loss (15) i and according to (12), the boundary of Zi is de ned by a set f (i; i) j Pr[ qi  minfi; (Di ? Ai)ig j i = i; Ki = 1 ] = Pi g: (16) This set is illustrated by the thickened curve in Fig. 2b. Notice that reduction of buer space i can be achieved by increase of transmission bandwidth i, and vice versa, without violating QOS. i i are On the other hand, the individual bounds (13) and (14) are applied when P~blocking and P~delay separately controlled. Similarly, the following theorem states the end-to-end loss performance bounds for FRPS connections.

Theorem 2.

(1) (k) (k) If FRPS connection of session i is set up by ((1) i ; i ) = (i; i ), (i ; i ) = (i; 0), k 6= 1, P and j 2I m mj + mi < C m , m 2 Ri , then i  Pr[ qi  minfi; (Di ? Ai )ig j i = i; Ki = i ]; P~loss (17) i P~blocking  Pr[ qi = i j i = i; Ki = i ] (18) and ( i ? Ai )i  i i ~ Pdelay  0Pr[ qi  (Di ? Ai )i j i = i ; Ki = i ] ifif ((D (19) Di ? Ai )i > i. The proof is provided in Appendix. Similar to RPPS, the admissible set for FRPS session i is given by Zi = f (i ; i) j Pr[ qi  minfi; (Di ? Ai )ig j i = i; Ki = i ]  Pi g: (20) Compared to the RPPS discipline, the FRPS discipline has following three advantages. First, for the same (i ; i), the FRPS connection has tighter loss performance bounds than the RPPS connection. This is because the bounds for FRPS are derived from queueing analysis of a nitebuer WC system (Ki = i ) whereas the bounds for RPPS are from that of an in nite-buer WC system (Ki = 1) (refer to Proposition). Second, the FRPS connection requires i amount of buer space only at the rst node, whereas the RPPS connection requires the same buer space at every node in the route. Third, the best-eort trac will achieve better performance under the FRPS discipline. This is because any unused bandwidth of each individual FRPS connection will be instantaneously taken by best-eort trac, whereas best-eort trac under the RPPS discipline can be transmitted if and only if all the connections are simultaneously idle. In other words, the RPPS connections always have high priority to transmit over the best-eort trac, which is not true with the FRPS connections. For the same reason, the actual performance of the RPPS connections (i.e., not the bound performance) should be better than that of the FRPS connections, even though the two disciplines provide similar bound performance.

6

3

-Characterization of Multimedia Trac

(; )

Although (; ) representation has been used to describe either an original session [3, 7, 9] or a leakybucket constrained session [5], the actual relationship between (; ) and real trac characteristics has not been explored. The emphasis in this section is placed on (; )-characterization of several representative multimedia trac streams. Notice that (; )-characterization of each individual session requires the only analysis of a WC queueing system of its own session, completely separate from the network environment and any other sessions. For deterministic (; )-characterization, the trac rate function xi (t) is deterministic. When xi(t) is applied to an in nite-buer WC system with transmission bandwidth i, the queueing process must also be deterministic and evolved by

qi(t + 1 ) = [ qi (t) ? 1 ]+ + i

Z t+ 1 i

t

xi ( )d:

In practice, the deterministic (; )-characterization is suitable for stored media communications such as video-on-demand (VOD) whose rate function xi (t) can be determined a priori. For probabilistic (; )-characterization, the trac is stochastic and so is the queueing process. Based on long-term statistical measurement, each typical multimedia trac can be described by two statistical functions: rate distribution f (x) and power spectrum P (! ). The study in [14, 15] shows that queueing performance is mainly determined by the two input statistical functions. Here, we use a sophisticated statistical match queueing (SMAQ) tool [15] to construct a multi-state CMPP whose statistical functions match with f (x) and P (! ). As shown in [15], one can use a CMPP to match with a wide range of the statistical functions. In contrast, using the superposition of two-state MMPP's as commonly adopted in trac modeling, P (! ) has to be a monotone function of j! j and f (x) is limited to convolution of mixed rate binomial functions. The queueing process of a WC system with CMPP input can be modeled as a quasi-birth-death (QBD) process [16]. In this paper, we use the so-called QBD folding algorithm [17] to compute the queue distribution function Pr[ qi  i j i = i; Ki ]. Since Ki must be nite for the folding algorithm, in numerical study one can always take Ki suciently large to approximate an in nite-buer WC system. Other queueing techniques, such as the matrix-geometric solution approach [16], can also be applied for the computation. Unlike the deterministic (; )-characterization, the probabilistic (; )-characterization can be applied to live media communications as long as its trac statistics can be collected a priori.

A. MPEG/JPEG Video Trac Take a 3.5-minute segment of the movie Star Wars, which is encoded into MPEG and JPEG video sequences [19]. Both sequences are recorded in cells per frame (cpf), and there are 24 frames per second. In the MPEG coding, we use predictive motion compensation only and take the ratio of number of I-frames to P-frames equal to 15 [18]. The average rate of the MPEG sequence is 227.7 cpf (2.3 Mbps) and that of the JPEG sequence is 345.3 cpf (3.5 Mbps). Fig. 3a compares the deterministic (; )-characterization of the two video sequences. Given at the same transmission bandwidth i, the JPEG video obviously requires substantially more buer space i than the MPEG video in order to avoid any cell loss, except at higher bandwidth. 7

1e4

JPEG

D i - A i = 480

i

6e3

σ (cells)

4e5

6e5

2e3

MPEG

D i - A i = 30

0

0

2e5

i

σ (cells)

JPEG

MPEG

300

400

500

ρi (cpf)

600

300

400

500

ρi (cpf)

600

(b)

(a)

Figure 3: (a) Deterministic (; )-characterizations of MPEG/JPEG videos (b) end-to-end queueing delay constraints on (; )-characterization. The end-to-end delay constraint, Di , for video services is typically in the range of 50  500 msec depending on applications [20]. Assuming Ai = 20 msec, the end-to-end queueing delay constraint is given by Di ? Ai = 30  480 msec. The dotted lines in Fig. 3b show such constraints imposed on the (; )-characterization curves. Taking the maximum queueing delay constraint Di ? Ai = 480 msec, for example, the bandwidth i is minimized to 350 cpf ( 3.6 Mbps) for the MPEG video and 580 cpf ( 5.9 Mbps) for the JPEG video. The corresponding maximum buer space i is about 4,000 cells and 6,000 cells, respectively. By comparison, the MPEG coding with motion compensation can save considerable transmission bandwidth and buer space in video transmission. A similar comparison can be made in Fig. 3b for the stringent delay constraint Di ? Ai = 30 msec, except that more bandwidth is required to trade for less buer space as Di ? Ai reduces. The reason for the MPEG video to require less network resources is well explained from the statistics of the two video sequences. First, we compare the rst-order statistics (histogram) of the two sequences by plotting the corresponding rate cumulative distribution functions (CDF) in Figs. 5a and 6a. Obviously, the MPEG CDF concentrates more on lower input rate than the JPEG CDF. Second, we inspect the second-order statistics (power spectrum) of the two sequences in Figs. 5b and 6b. It is clear that the MPEG video has much less power in the low-frequency band than the JPEG video. The less the low frequency power, the better the queueing performance will be [14]. Next, we study the eect of trac smoothing on the (; )-characterization of MPEG video. In general, I -frame appears periodically in an MPEG video stream, such as at the every interval of 16 frames in our example. Also because of motion compensation, B - and P -frames, which appear in adjacent to I -frames, consist of much less number of cells than I -frames. As a result, the generation of I -frames corresponds to periodic burst arrivals in an MPEG video stream. To prevent nodal congestion, many control protocols have been proposed to smooth such bursty trac streams at user network interface (UNI) [22]-[24]. Naturally, adding a trac smoothing device will introduce extra queueing delay at the source and/or at the destination during playback. For example, consider a simple averaging device for smoothing [24]. De ne the averaging time-interval by T . The device holds cells arriving in the time interval [(n ? 1)T; nT ) at a smoothing buer, computes their average arrival rate, and then release them to the network during the next time interval [nT; (n + 1)T ) at that average rate. The average queueing delay introduced by this device is assumed to be approximately T . Fig. 4a shows the impact of smoothing on the deterministic (; )-characterization of the MPEG video. The solid curve is for no smoothing, i.e. T = 0; the 8

10

15

0

0

5

PSD (x105 )

4e3 2e3

σ i (cells)

6e3

T=0 T=4 T=8

300

350

400

ρi (cpf)

450

500

0

10

20

30

40

50

60

70

radian frequency ω

(a)

(b)

Figure 4: (a) Eect of trac smoothing on (; )-characterization of MPEG video (b) power spectrum of MPEG video. non-uniformly broken curve is for smoothing at T = 4 frames; and the dotted curve is for smoothing at T = 8 frames. Obviously, the (; )-characterization of MPEG video is basically unchanged by smoothing (unless T is unreasonably large). In other words, the network resource saving by input smoothing for MPEG video is insigni cant! Moreover, such a smoothing introduces extra delay of 167 msec at T = 4 frames or 333 msec at T = 8 frames. The end-to-end network queueing delay constraint, Di ? Ai , therefore becomes much stringent as illustrated in Fig. 4a. In consequence, the minimal bandwidth for QOS guarantee needs to be substantially increased with T . The admissible set Zi also becomes smaller as T increases. The ineectiveness of smoothing at T = 4, 8 frames can be explained from the MPEG video power spectrum in Fig. 4b. The spectral spikes appeared at harmonic radian frequencies 24 16  2k, k = 1; 2;


, are attributed to periodicity of the I -frames. Yet, a large amount of video power is located in a low-frequency band, typically ! < 2 radians. In signal processing context, smoothing at T sec is somewhat equivalent to low-pass ltering at cuto frequency !c = 2T radians. Obviously, the low-frequency video power cannot be ltered out by smoothing unless T can be much longer than a second. On the other hand, the queueing performance of the MPEG video is mainly determined by the video statistics in the low-frequency band (i.e., scene changes) [12]. This is why the MPEG video (; )-characterization is basically unaected by the smoothing in Fig. 4a. For probabilistic (; )-characterization, we use the SMAQ tool [15] to construct a 101-state CMPP model to match with important statistics of the 3.5-minute MPEG/JPEG video sequences. Figs. 5a and 5b show the comparison of the CDF and power spectrum between the CMPP model and the real MPEG video sequence. The CDF of the CMPP model matches exactly with that of the real sequence. In power spectrum matching, emphasis is placed on the low-frequency band, whereas the spectral spikes, which are attributed to periodic bursty I -frames, were ignored because the queueing performance is mainly aected by the low-frequency power. Note that the logarithmic scale is used in Fig. 5b for the power spectrum. To check the validity of the model, we compare the analytical queueing solution of the CMPP to the simulated queueing solution of the nite real MPEG video sequence in a WC system. The solutions of mean queue length, queue standard deviation and average loss rate are compared in Figs. 5c, d, assuming a nite buer capacity of 1,000 cells. As one can see, the model-based analytical queueing solutions are suciently close to the simulated queueing solutions at dierent utilizations. Similarly, we constructed a 101-state CMPP model for the JPEG video, as the results are shown in Fig. 6. Using the above two CMPP models, one can derive the probabilistic (; )-characterization of 9

e+1 PSD (log10)

CDF

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

_____ measured - - - - - CMPP 1

2

3

4

5

_____ measured - - - - - CMPP

e+0

e-1

e-2 e-3

6

0

5

bit rate (Mbps) (a)

10 15 20 radian frequency ω

25

(b)

450 0.12

stand. devi.

350

loss rate

queue (cells)

400

300

mean 250 200

100 0.65

0.04

_____ CMPP - - - - -simulation

150 0.7

0.75

0.8

0.85

0.9

0.08

0 0.65

0.95

utilization (c)

_____ CMPP - - - - -simulation 0.7

0.75

0.8

0.85

0.9

0.95

utilization (d)

Figure 5: MPEG Star Wars modeling by CMPP and queueing performance (a) rate cumulative distribution (b) power spectrum (c) mean and standard deviation of queue (d) loss rate. the MPEG/JPEG Star Wars, as plotted in Fig. 7. The inspection of Figs. 7a, c indicates that for each given i , the solution Pr[ qi  i j i = i; Ki = 1 ] of the MPEG video is much less than that of the JPEG video at each given i . It implies that the MPEG video requires much less buer space than the JPEG video at each given bandwidth. Similarly, as shown in Figs. 7b, d, the MPEG video requires much less bandwidth than the JPEG video at each given buer space. Another interesting observation is that the function Pr[ qi  i j i = i; Ki = 1 ] decreases almost exponentially with respect to i at each given i. In contrast, it decreases faster than exponential with respect to i at each given i .

B. Voice Trac A two-state MMPP, alternating between ON- and OFF- states, is typically used to model a voice trac. De ne a two-state MMPP by "

#

Q = ? ?  ; ~ = [ 0; on ];

(21)

where on is the Poisson rate while in ON-state. Then, = +  is the steady-state probability of ON-state and = on is the average rate. Consider a 64 Kbps PCM voice source where the average silent and talkspurt periods are equal to 0.6 sec and 0.4 sec, respectively. Based on the ATM cell packetization, we get ( on , ?1 , ?1 ) = (151 cells/sec, 0.6 sec, 0.4 sec), and so  = 0.4 and  = 60.4 cells/sec. The probabilistic (; )-characterization of each voice source is plotted in Fig. 8a. The end-to-end cell loss constraint Pi for voice services is dependent on the coding and priority 10

e+2

0.8 0.7 0.6

e+1

PSD (log10)

CDF

1 0.9

0.5 0.4 0.3 0.2 0.1 0 1

_____ measured - - - - - CMPP

e+0 e-1

_____ measured e-2

- - - - - CMPP 2

3 4 5 bit rate (Mbps)

6

e-3 0

7

5

10 15 20 radian frequency ω

(b)

(a) 0.1 0.08

stand. devi.

300

mean

250 200 150 100 50 0 0.65

loss rate

queue (cells)

500 450 400 350

25

_____ CMPP - - - - -simulation 0.7

0.75

0.8

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_____ CMPP - - - - -simulation

0.02 0 0.65 0.7

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utilization (c)

0.75

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0.9 0.95

utilization (d)

Figure 6: JPEG Star Wars modeling by CMPP and queueing performance (a) rate cumulative distribution (b) power spectrum (c) mean and standard deviation of queue (d) loss rate. packetization techniques. For the plain PCM coding without priority packetization, one can choose Pi = 10?2 [21]. Let us also bound the end-to-end delay of voice by 50 msec and so Di ? Ai = 30 msec. In Fig. 8a, one may nd an empty admissible set Zi if the bandwidth is limited by i  62 Kbps. In other words, under the condition of Di ? Ai = 30 msec and Pi = 10?2 , there will be no solution (i ; i ) to satisfy Pr[ qi  minfi ; (Di ? Ai )i g j i = i ; Ki = 1 ]  Pi , if i  62 Kbps. Note that the maximum voice bandwidth is only 64 Kbps at which we can have i = 0 as in the circuit-switched case. Thus, virtually no improvement can be achieved by packet switching over circuit switching for voice transmission! This problem is essentially caused by the \narrow" bandwidth of voice and its \stringent" delay constraint. Obviously, even holding a few cells in the buer would result in delay exceeding 30 msec in a voice connection with bandwidth less than 62 Kbps. This is an inherent problem with the (; )-characterization since the analysis of WC system is based on each individual session at its own transmission bandwidth, no matter how many sessions are multiplexed and how large the aggregate transmission bandwidth is on each link. One possible solution is to group a large number of small sessions into a \super" session at UNI, as to be discussed in Section 4. We now examine the impact of voice ON/OFF periods on the (; )-characterization. Let the average ON- and OFF-periods be simultaneously scaled, which will change the voice power spectrum but not its rate distribution [11]. When (?1 , ?1 ) is increased to (1.2 sec, 0.8 sec), more voice power concentrates on the low-frequency band and so the queueing performance deteriorates, and vice versa when (?1 , ?1 ) is reduced to (0.3 sec, 0.2 sec). Fig. 8b shows the corresponding (; )-characterizations as a function of i at i = 200 cells. Obviously, the voice transmission bandwidth i must be increased with the length of ON- and OFF-periods for the same value of 11

1

1

σi 1e3

e-4

2e3 3e3

e-6

4e3 e-8

3.8

4

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e-4

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1

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σ i (cells)

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σi

5 6e3 7e3

ρi

e-2 1e3

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e-8

Pr [qi > σ i | ρi ]

Pr [qi > σ i | ρi ]

4e3

(b)

(a)

e-2

3.8 4 4.2 4.4

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5e3 e-10

ρi

e-2

Pr [qi > σ i | ρi ]

Pr [qi > σ i | ρi ]

e-2

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4.9 5.1 5.3 5.5 5.7

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5.2 5.4 5.6 5.8

ρi (Mbps)

6

6e3 6.2 6.4

e-10

1e3

2e3

3e3

4e3

5e3

σ i (cells)

6.1 6e3 7e3

(d)

(c)

Figure 7: Probabilistic (; )-characterization of Star Wars (a)(b) MPEG case (c)(d)JPEG case. Pr[ qi  i j i = i; Ki = 1 ].

C. LAN-to-LAN Data Trac Consider a scenario where two geographically distributed Ethernets are connected via an ATM backbone network. Assume that the whole LAN-to-LAN data trac in each direction is transmitted as one session. The trac modeling is based on the statistics of a 3.5-minute Ethernet data trace collected on an Ethernet cable at Bellcore [25]. The average input rate is about 2.0 Mbps. Refer to the detail construction of CMPP for the Ethernet trac in [15]. Its probabilistic (; )characterization can then be found in Fig. 9.

4 Admission Control and Resource Sharing When a deterministic session i arrives, its deterministic (; )-characterization and end-to-end delay constraint Di will be provided to the network. When a probabilistic session i arrives, its trac type, delay constraint Di and loss constraint Pi will be provided to the network. For each given trac type, the probabilistic (; )-characterization can be identi ed from network database. Building such a database in advance is feasible based on a WC system queueing analysis, once the statistical functions f (x) and P (w) of certain representative trac streams are collected. The function of admission control is rst to determine an admissible set Zi , and then to select a proper (i ; i ) 2 Zi according to present network-wide resource availability. If none of the (i ; i)'s in Zi can be ensured by the network, the arriving session will be blocked. 12

1 e-2

Pr [qi > σ i | ρi ]

Pr [qi > σ i | ρi ]

1 e-4 e-8 e-12 0

σ 100 i (c e

e-6 ( α -1,

e-8

)

40

30

60

50

e-14 25 30

ρ i (Kbps)

β -1)

( 1.2, 0.8 ) ( 0.6, 0.4 ) ( 0.3, 0.2 )

e-10 e-12

200

lls

e-4

35

40

(a)

45

50

ρ i (Kbps)

55

60 65

(b)

Figure 8: Probabilistic (; )-characterization of voice trac (a) when ( on , ?1 , ?1 )=(151 cells/sec, 0.6 sec, 0.4 sec) (b) impact of ON/OFF period changes at i = 200 (cells). 1

σi

e-2

600 e-4

1000

e-6

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e-8

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Pr [qi > σ i | ρi ]

Pr [qi > σ i | ρi ]

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1 200

3.6 e-4

4.1

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3

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ρi (Mbps)

e-12 0

6

3.1

(a)

1e3

2e3

σ i (cells)

3e3

(b)

Figure 9: Probabilistic (; )-characterization of LAN-to-LAN data trac driven by an Ethernet. The admission rule for RPPS connections is de ned as follows. When session i requests a connection, if the network resource manager can identify a (i; i) 2 Zi and nd a route Ri such that X m m and i + X m < C m ; m 2 Ri ; i + (22) j B j j 2I m

j 2I m

the connection will be established with ( im; mi ) = (i ; i ), m 2 Ri . The tradeo between buer space and bandwidth is examined in the identi cation of (i ; i ) to comply with present network resource conditions. The admission rule is equally applied to deterministic and probabilistic connections. Once probabilistic session i is admitted under the rule, its QOS is guaranteed without violating the QOS guarantees of on-going sessions (refer to Theorem 1). On the other hand, if session i requests a FRPS connection, it is sucient to examine the buer space condition in (22) for the rst node in route Ri . The QOS of each probabilistic FRPS connection is thus guaranteed from Theorem 2. The importance of the admissible set Zi is to provide a great exibility in connection management. According to present network-wide resource availability, the network manager can properly adapt the (; ) parameters of all sessions, as long as the (; ) of each connection is selected from its own admissible set. It also plays an important role in resolving the so-called network resource fragmentation. When a new session cannot be accepted because of resource fragmentation, the network can selectively re-negotiate the connection parameters of on-going sessions to \create" resources for the new session. 13

4.1 Buer Sharing One de ciency of the above admission rule is that resource sharing is completely neglected. The buer space allocation policy in (22) assumes that a separate buer space i is dedicated to each connection at every node. Similarly, the bandwidth allocation policy in (22) does not take advantage of statistical multiplexing gain, i.e., the direct sum of connection bandwidths at each node never exceeds the link capacity. This is also why the end-to-end per-session performance bound can be determined from queueing analysis of an \isolated" WC system, independent of the network environment and other connections. The number of connections that can be supported in a network can be substantially reduced by the admission rule (22). For the probabilistic connections, it is possible to take advantage of buer sharing in the design of admission control. Instead of dedicating a segregate buer space to each connection, one can make the overall buer space at each node to be statistically shared by all the connections. By the law of large numbers, the total buer space requirement can be substantially reduced without violating the QOS guarantee of each individual connection. Let x(i k) (t) be the arrival rate of session i at time t on the k-th link. The aggregate arrival rate on the k-th link is denoted by x(k)(t). When the k-th buer is full, the aggregate loss rate is equal to l(k) (t) = [ x(k) (t) ? C (k) ]+ according to the uid ow analysis. The amount l(k) (t) can be arbitrarily distributed among all the arrival rates x(i k) (t), 8i, through the implementation of cell selective discarding. In our case, we assume a rate proportional (RP) loss distribution. Denote the loss rate of session i at time t by li(k)(t). The RP loss distribution ensures (k) li(k)(t) = xx(ik)((tt)) l(k)(t); 8 i 2 I (k):

(23)

The instantaneous loss ratio of session i is then given by

li(k)(t) = [x(k)(t) ? C (k) ]+ ; 8 i 2 I (k); (24) x(k) (t) x(ik) (t) which is identical for all the sessions at the k-th node during the buer blocking period. De ne the steady state loss probability of session i at the k-th node, L(i k) , by the expectation of its instantaneous loss ratio, 8i. From (24) we get L(i k) = L(jk), i; j 2 I (k) , i.e., the loss probability of each session at the k-th node must be identical under the RP loss distribution.

In practice, the RP loss distribution is naturally implemented without control of cell selective discarding. That is, when the buer is full, the instantaneous loss rate of each session is proportional to its own instantaneous arrival rate. The following theorem states the end-to-end loss performance bounds for RPPS connections with buer sharing. Theorem 3. Suppose that each RPPS connection is established with (ik) = i and Pj2I k (jk) < C (k) , (k) 2 Ri, and the buer at each node in the route is statistically shared by other connections. Then, under RP loss distribution, ( )

i P~blocking  1?

Y (k)2Ri

( 1?

Z1

B (k)

14

[ j 2I k hj (qj ; j ) ] dq (k) ) ( )

(25)

and i P~delay 

(

P

Pr[ qi  (Di ? Ai )i j i = i ; Ki = 1 ] if (Di ? Ai )i  P(k)2Ri B (k) 0 if (Di ? Ai )i > (k)2Ri B (k) P

(26)

where is a convolution operator and q (k) = j 2I k qj is the aggregate queue length at the kth buer of connection i. hi (qi ; i) is the probability density function (PDF) of qi when session i is applied to an \isolated" WC system with i = i and Ki = 1. By de nition, hi (qi; i) = d dqi Pr[ qi  i j i = i; Ki = 1 ]. The proof is provided in Appendix. i The bound on P~delay is solely determined by the (; )-characterization of session i, whereas i ~ the bound on Pblocking requires the (; )-characterization of all the interacting connections in the route. R For convenience, denote j 2I m hj (qj ; j ) and B1m [ j 2I m hj (qj ; j ) ]dq m by hm and BLKm , respectively. Assume that the network manager keeps the present information of hm and BLKm for j , denoted by DLY , for all shared buers in the network. Also, it keeps the present bound on P~delay j all connections in the network. The admission control for RPPS connections with buer sharing is de ned as follows. When session i requests a RPPS connection, the network manager selects candidate i and Ri , and update BLK(k)

=

Z1

B (k)

( )

[ h(k) hi (qi ; i) ] d(q (k) + qi );

8 (k) 2 Ri:

(27)

Then, the following conditions are sucient to provide the guaranteed QOS of session i without violating the QOS of on-going sessions: X (k) i + j < C (k); 8 (k) 2 Ri ; (28) 8 > < > :

and

Q

j 2I (k)

(k)2Ri (1 ? BLK

(k)) + Pr[

qi  (Di ? Ai )i j i = i; Ki = 1 ] PPi; if (Di ? Ai )i  P(k)2Ri B (k) ; Q 1 ? (k)2Ri (1 ? BLK(k))  Pi ; if (Di ? Ai )i > (k)2Ri B (k) ;

1?

1?

Y (l)2Rj

(29)

(1 ? BLK(l)) + DLY j  Pj ; 8 j 2 i :

(30)

i is the set of all the connections interacting with the new connection i, which is de ned by i = f j j Rj \ Ri 6= ; g. (28) examines the bandwidth availability along the route, (29) ensures j j i i  Pj , j 2 i . + P~delay P~blocking + P~delay  Pi for the arriving session i, and (30) guarantees P~blocking Similarly, the following theorem states the end-to-end loss performance bounds for FRPS connections with buer sharing. Theorem 4. Suppose that each FRPS connection is established with (ik) = i and Pj2I k (jk) < C (k) , (k) 2 Ri , and the buer space at each node in the route is statistically shared by other connections. Then, under RP loss distribution, Z1 i ~ (31) [ j 2I hj (qj ; j ) ] dq (1) Pblocking  ( )

and

(1)

B (1)

i P~delay 

(

Pr[ qi  (Di ? Ai )i j i = i; Ki = 1 ] if (Di ? Ai )i  B (1) 0 if (Di ? Ai )i > B (1) . 15

(32)

30

250

25 20 Nσ i σ sharing

N = 100

200

MPEG JPEG

σ super

15

(cells)

150

N = 50

100

10 50

5 0 0

5 10 15 20 25 30 35 40 45 50

N

0 25

N = 10 30

35

40

45

N (b)

(a)

50

ρ super

55

60

65

(Kbps)

Figure 10: (a) Reduction of buer space requirement by sharing (b) reduction of bandwidth requirement by trac aggregation. The proof is provided in Appendix. i Compared to RPPS, FRPS provides a tighter bound on P~blocking since buer sharing is required only at the rst node in Ri . Similar admission control can be implemented for FRPS connections. It is sucient to examine the conditions (28), (29) and (30) at the rst node. The admission control with buer sharing will greatly improve buer resource utilization, yet it requires considerable increase of computational overhead for connection management. This is because the (; )-characterizations of all interacting connections must be taken into account for the evaluation of per-session loss performance bound. Let us investigate the reduction of buer space requirement by sharing. For simplicity, consider N sessions of i.i.d. MPEG/JPEG video sources which are routed on a network link. In terms of bandwidth allocation, each session requires the same amount of bandwidth i ; the aggregate bandwidth N  i must be less than the link capacity. Without buer sharing, each session requires a separate buer space allocation i ; the aggregate buer space requirement is N  i . For instance, take the loss probability bound Pr[qi  i ji = i ; Ki = 1] = 10?6 in (13). One can then get i = 4,689 cells at i = 4.8 Mbps for each MPEG video session in Fig. 7b and i = 5,106 cells at i = 5.9 Mbps for each JPEG video session in Fig. 7d. When the buer sharing is considered by the admission control, denote the aggregate buer space requirement by sharing , subject to the same loss probability bound 10?6 per session under the RP loss distribution. Taking the N -th convolution of the video (; )-characterization, one can nd sharing from the bound solution in (25). Fig. 10a shows the ratio of Ni to sharing as a function of N . At N = 50, only less than 5% of total buer capacity is required by sharing for the MPEG/JPEG video using the same transmission bandwidth! Obviously, sharing can save substantial amount of buer space. In the admission control, one can always trade more buer resource i for less bandwidth i since the shared buer space will not be signi cantly increased by the individual i .

4.2 Trac Aggregation In this subsection, we investigate the eect of trac aggregation on bandwidth saving. In practice, when several sessions at UNI share a common route, they can be grouped into a \super" session. Such a grouping has the same eect of statistical multiplexing on bandwidth eciency. For example, 16

consider the existence of a large number of voice sessions between two distant local switching centers which are interconnected via a backbone ATM network. Instead of frequently setting up each individual voice connection on per-session basis, one can infrequently set up a \super" connection between the two switching centers to transport multiple voice sessions. For simplicity, assume that each voice session is a 64 Kbps PCM voice with silence detection, which is modeled by the two-state MMPP in Section 3B. As previously discussed in Section 3, under the constraints Di ? Ai = 30 msec and Pi = 10?2, the transmission bandwidth of each voice connection without grouping must be greater than 62 Kbps. For N voice sessions, if each session is set up as a separate connection, the total amount of bandwidth must be greater than N  62 Kbps. In contrast, when all the N voice sessions are grouped into a \super" connection, the aggregate bandwidth requirement can be substantially reduced as indicated Fig. 10b. The curves represent the boundary of the set of admissible (super ; super ) for the \super" connection with respect to N = 10, 50, 100, subject to Di ? Ai = 30 msec and Pi = 10?2 . (super ; super ) represents the buer space and bandwidth requirement for the \super" connection. The bandwidth per voice session is measured by super N , which can be greatly reduced as N increases.

5 Conclusion This paper has presented a methodology to establish real-time connections with guaranteed QOS for stochastic sessions. Once network resources are allocated to each session according to its (; )characterization, one can probabilistically bound per-session end-to-end loss performance. Upon request of each new connection, the network manager determines the admissible set Zi based on its (; )-characterization and QOS requirement, and then selects a proper (i ; i ) 2 Zi according to present network-wide resource availability. In order to improve network resource utilization, we have also studied the admission control with buer sharing and statistical multiplexing, subject to the same QOS guarantee of each connection. The detailed study on the (; )-characterization of real multimedia trac has revealed the interrelationship between (; )-characterization and trac statistics and also provided many engineering aspects of (; )-characterization for connection management.

APPENDIX Proof of Proposition. Compare the queueing process of a WC system with three dierent buer capacities, Ki = i , i  Ki < 1 and Ki = 1, for the same realization xi (t) of session i. A time interval of qi (t) > 0 is called busy period. Consider a WC system at Ki =R i with its busy periodR denoted by t 2 [s; e]. During the busy period, we can write qi (t)jKi =i = st xi ( )d ? i (t ? s) ? st li( )jKi=i d , where li (t)jKi=i is the loss rate at time t. Next, we consider the same WC system except with i  Ki < 1 and its busy period is denoted by [s; e0 ]. Obviously, we must have e  e0 since the busy periodR of the former system must belong to that of the latter system. One Rt t can then write qi (tR)ji Ki