A two-layer congestion control protocol for broadband ISDN

Performance Evaluation 16 (1992) 85-106 North-Holland

85

A two-layer congestion control protocol for broadband ISDN * Zhixing Ren Cogent Data Technologies, 175 West Street, P.O. Box 926, Friday Harbor, WA 98250, USA

James S. Meditch Department of Electrical Engineering, FT-IO, University of Washington, Seattle, WA 98195, USA

Abstract Ren, Z. and J.S. Meditch, A two-layer congestion control protocol for broadband ISDN, Performance Evaluation 16 (1992) 85-106. in this paper, we present a new, effective connection-oriented congestion control protocol for B-ISDN. The protocol employs the traffic viewpoint hierarchical design approach and is implemented by a two-layer scheme at the call and cell layers. The call layer handles call admission (denial) and provides shortest path connection routing when a call is admitted, while the cell layer allocates switch input buffer space for each traffic type according to each type's cell loss probability grade-of-service requirement. There is a direct interaction between the two layers in processing a connection request. A discrete-time queueing system with geometrically distributed service time and state-dependent Markov modulated Bernoulli process (MMBP) arrivals is established to model heterogeneous networking environments for this protocol and an analytical sulutioa i~ developed for this queueing system. Numerical results obtained by both analysis and simulation show that the protocol can reduce network congestion as input traffic rates approach switch capacity and, thereby, increase network utilization relative to that possible in the absence of congestion control.

Keywords: broadband ISDN; congestion control; multi-layer protocol; traffic models; matrix geometric solutions.

1. Introduction

The asynchronous transfer mode (ATM) has emerged as the transport method for the broadband integrated services digital network (B-ISDN). The strength of ATM is derived from its unified transport structure and its ability to support multirate services ranging from low speed telemetry to simultaneous multimedia services requiring text, image, video and voice, while also supporting a variety of other point-to-point and muitipoint voice, data and video services. In order to realize flexible, fair and efficient sharing of network resources by users requiring muitirate services, ATM-based B-ISDN's must take advantage of statistical multiplexing and deliver information with its respective grade-of-service (GOS) requirements, viz., cell delay and cell loss, to the intended users. In general, maximizing the utilization of network resources while satisfying GOS requirements in the presence of traffic congestion is the goal of congestion control mechanisms. Specifically, the major functions of congestion control for B-ISDN are: (!) prevention of throughput degradation and inefficiency due to network overload, (2) reduction of packet loss and delay, and (3) fair allocation of network resources among competing users when traffic congestion occurs. Some significant differences be~een traditional data networks and ATM networks should be taken into account in designing congestion Correspondence to: Dr. Z. Ren, Cogent Data Technologies, 175 West Street, P.O. Box 926, Friday Harbor, WA 98250, USA. * This research was supported in part by the National Science Foundation under NSF Grant NCR-913485. 0166-5316/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

control protocols for B-ISDN. First, while traditional data networks do not generally carry real-time traffic, B-ISDN supports both real-time and non-real-time traffic over a range of applications requiring bandwidth, equivalently, bit-rates, from a few Kbits/s to hundreds of Mbits/s. In addition, different services require different GOS, e.g., delay sensitive service for video and voice, loss sensitive service for data [24]. Second, switching and transmission in B-ISDN are carried out at very high spc::d, making the network t,olatile in terms of traffic control. As a result of the drastically reduced packet transmission times relative to the intrinsic propagation delay, the use of feedback control schemes, such as window control schemes in current packet-switching networks, is inappropriate for B-ISDN. An important issue in the study of congestion control figr B-ISDN is that of modeling for performance analysis. Most switching system modeling and performance analyses are based on two assumptions. One is that packet arrivals at each input are defined by an independent Bernoulli process [8,11,12], the other assumes that the destination address of each packet is independently and uniformly assigned. The first assumption is not valid in the case of B-ISDN since the traffic is usually bursty and correlated, an example being a data burst during a video teleconference. As is well known, queueing systems with correlated inputs are far less well understood than those with independent inputs [15,20,21,29,30], and analytical solutions are, in general, difficult to obtain. Therefore, most performance studies in the past have either been restricted to oversimplified models or relied on simulation. The design of a congestion control protocol may be envisioned as a system design problem which is structured into a multilayer spatial hierarchy to simplify the design procedure. Two viewpoints on layered design models have appeared in the recent past. The first of these, termed the network viewpoint model [6], consists of a service layer, an ATM adaptation layer, an ATM transport layer and a physical layer. The ATM transport layer may be divided further into a network layer and a link layer, and traffic control may be imposed on entities at these layers. The second, which is called the traffic viewpoint model, is based on the observation that the traffic flow in an ATM network has a common, multilayer structure, even though the traffic flow is a mixture of cells from a variety of users and applications. As shown in Fig. 1, calls are composed of bursts which arc, in turn, composed of cells (or packets) [5,10]. The resulting model is then dc(ined by a call layer, a burst layer and a cell layer, respectively. An advantage of this formulation is that the resultir.~ control scheme will be suitable for all traffic types in heterogeneous networking environments because all sources have the same underlying characteristics when they are presented at the same layer.

Zhixing Ren received the B.S. and M.S. degree in engineering from the Changsha Institute of Technology, China, in 1982 and 1984, respectively, and the M.S. and Ph.D. degree in electrical engineering from the University of Washington in 1989 and 1991, respectively. In August 1991, he joined Cogent Data Technologies, Friday Harbor, WA, where he is currently an R&D engineer. Dr. Ren's research interests include telecommunication switching, broadband network architecture and protocols, computer communication system design, modeling, simulation and performance analysis.

James S. Meditch received his undergraduate and gradua,e education in electrical engineering at Purdue and M.I.T. His career includes 8 years of industrial R&D experience, and 23 years of university teaching, leseareh, and administrative responsibilities. Throughout ilis academic career, he has also served as a consultant to government and industry. He has been with the University of Washington since 1977. Professor Meditch's current research interests include communications switching and traffic theory, broadband integrated services telecommunication networks, multimedia communications, network performance modeling, analysis and optimization, and broadband network architectures and protocols.

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

]-" callholdi~,g ~ time i . . ~ '~

]

87

CALLiAYER

callinter-arrivaltime ,silence period • i ~ cell length

BURSTLAYER~

burstduration CELLLAYER

nnnnnn

nnnnnnn

cell inler-~rival time

Fig. 1. Multilayer structure of traffic sources.

A number of congestion control protocols have been proposed in [1,2,9,12,28,31-33]. Turner [31] suggested a connection-oriented protocol for B-ISDN. In his approach, a connection must be setup before communication takes place. Th? lin~ layer then employs a leaky bucket scheme, which is based on the use of counters, to filter traffic flow. To prevent or limit congestion in the network, packets are leaked, i.e., discarded when the packet counter reaches a pre-determined threshold. Woodruff and Kositpaiboon [32] argued that traffic flow in ATM networks should be controlled on an end-to-end basis instead of link-to-link in order to reduce processing overhead at intermediate nodes and to increase transmission speed within the network. They suggested employing reactive or preventive control schemes at network access interfaces to regulate traffic flow. Bala et al. [2] proposed a leaky bucket network access control scheme and studied its performance in an ATM environment using the matrix analytical approach [1]. They found that this end-to-end open loop control scheme is effective in fast packet-switching networks. Hui [10] and Filipiak [5] have described traffic viewpoint hierarchical designs in which traffic enforcement is carried out at the cell, burst and call layers, respectively. Bandwidth clipping and priority techniques are also considered viable link layer schemes for B-ISDN congestion control. Selective packet discarding [33], bit dropping [28], channel dropping [9] and group switching [12] are examples of these protocols. In this paper, we propose a connection-oriented congestion control protocol for B-ISDN. Three reasons for using this type of protocol are that it: • Provides high performance communications by separating more complex processing from data transfer, allowing simple hardware implementations, and reducing the data transfer overhead, especially for real-time video packets in switches and network interfaces. • Allows the network to make explicit resource allocation decisions when connections are established, thereby providing more predictable performance than is possible for connectionless networks. Hence, it can support real-time applications such as vnice conferencing and entertainment video with high quality. • Facilitates the support of multipoint communications. Connectionless protocols can support multipoint communications in one of two ways: transmission of packets with a list of destination addresses, or transmission to a multicast address. The first mechanism suffers large overhead in header processing when the number of end points is large. The second is actually using the notion of a connection, but lacks the flexibility and needs extra hardware to implement it. Our proposed congestion control protocol is based on the traffic viewpoint layered model and is implemented as a two-layer control via a call and cell layer scheme which is described in Section 2. In Section 3, we investigate the performance of our congestion control protocol in heterogeneous networking environments. Numerical examples are presented to demonstrate the performance of the protocol. Section 4 summarizes our contributions.

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Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

2. Protocol design and implementation In this section, we describe our congestion control protocol. As already noted, the protocol is based on the traffic viewpoint layered model and is implemented by a two-layer congestion control scheme. The call layer combines connection routing and call admission in optimally distributing traffic flow over the entire network while providing guaranteed GOS. The cell layer smooths the traffic burstiness which is due to large fluctuations in the traffic intensity of the various traffic types and allocates network resources to achieve the required service quality when congestion occurs. The objective of both two layers is to maximize network throughput and resource sharing while satisfying GOS requirements. Our congestion control protocol supports generic ATM networks. Figure 2 shows a simplified network diagram. The packet switches are composed of input processors, output processors, routing processors and a self-routing switch fabric. The input processor at each input port provides cell buffering, address mapping and congestion control enforcement and the output processor at each output port performs output synchronization and cell transmission. The routing processor handles call setup requests and provides connection routing. We assume that the switching fabric is internally non-blocking with speedup and does not buffer cells internally. The network interfaces include cell multiplexers and demultiplexers, and the associated user/network hardware and software systems to support end-to-end communications. To maintain high speed communications, end-to-end error detection and error recovery are assumed to be implemented at a higher layer. Given the above network architecture, we now define our two-layer congestion control protocol. We begin by considering a single link and define the link cost D ~ Pr(w > C) = B

(1)

where w denotes the multiplexed stream of the different traffic types on the link including the connection request that is being processed. In (1), the units on w and C are ceils/s; C is chosen to be less than the link's total capacity in order to provide a performance margin, e.g., C = 0.8 Cmax; and B is the value of the cell loss probability for the multiplexed stream. If w consists of only a single class of traffic whose cell loss requirement is R and B ~
P$: PacketSwitch

NI: NetworkInterface

Fig. 2. Network architecture.

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

89

average flow rate for class k traffic, k = 1, 2, 3. Under our approach to cell layer traffic enforcement, we allocate buffer space and enforce cell blocking and loss such that B2

B3

R~ - R 2

Bl

R3

~/~<

1

(2)

which ensures fairness in satisfying all cell loss GOS requirements. Letting A & ~ + ~22+ ~33,we see that the total average rate at which cells are lost is given by both BA and B I ~ + B2~22+ B3~33 via conservation of (loss) flow. Hence, BX - BiA'~l+ B2~2 + BaA'~3. From (2), B k - "fiR k, k ~- 1, 2, 3 and it follows directly using (3) that 'r/

(3)

AI+A2+A 3

=

+

+

(4)

Defining f k & B k / B _ r l R k / B ' k = 1, 2, 3 which are termed the cell layer control coefficients, we find from (4) that RkX f k _ R i l l + R2~2 + R3~3

(5)

and

Bk=fkB=Dk . (6) Hence, given B from (1) and invoking cell layer control according to (2), the actual cell loss probabilities for the three traffic classes are given by (5) and (6) for k ffi 1, 2, 3. From the above, which provides a procedure for establishing (or refusing) a virtual circuit connection between two adjacent nodes, we next consider a pair of source-destination nodes which are connected by one or more paths, and let j indexes paths while i indexes the links ,,long each path. Letting Di = P(wi > Ci) for link i, it follows under an independence assumption 1 on the w i that the path j which maximizes I=

i'-[ ( 1 - D i ) i Epathj

minimizes the cell loss probability of the connection. Equivalently, minimizing J-~ - I n I -

-

Y'. l n ( 1 - D i ) i ~path/

results in the same path. Since only cases where the D i are on the order 10 -3 or less are relevant, the approximation I n ( 1 - D~)= - D i can be used. We are thus led to the following optimization problem whose solution specifies our call admission and routing procedure: Given a pair of source-destination nodes and a set of associated paths {pathj}, find a path that minimizes J=

E

D,.

(7)

i ~path/

By a shortest path algorithm, the selected path satisfies J°&min{

~

D~}

path~ i E path j

and has the smallest achievable cell loss pr~aability for the new connection. If the cell loss GOS requirement for all traffic classes on each link of this path are satisfied, then the call is accepted. It is i This assumption is imuitively reasonable if the link is already carrying a large number of virtual circuit connections and the connection being considered for inclusion is a small fraction of the link's current load. We examine the validity of this assumption in our simulation studies for this case as well as others which differ significantly from this case.

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Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN queue length

7t3

~!~Sl~'

S~ontention

Fig. 3. Illustration of cell layer control scheme.

clear that the performance criterion J in (7) is related to real-time traffic conditions. When all connections are setup on the shortest paths, traffic flow in the network will be distributed optimally and network utilization will be maximized. We turn now to a description of our buffer allocation scheme for cell layer congestion control. We consider a large, internally nonblocking, space-division, time-slotted ATM switch with finite input buffers of size M, a single buffer at each output port for cell processing and transmission, and a switch fabric speedup capability of m >I 1. For the purpose of illustration, we consider three classes of traffic with R! >t R 2 >1 R 3, and average arrival rates AI, A2 and ~ , respectively. Figure 3 shows the diagram of the control scheme. The input buffer at each input port is a FCFS queue without priority and there are three thresholds K ! ~
3. Performance analysis We now analyze the performance of our proposed congestion control protocol in general heterogeneous networking environments. To begin, we consider traffic source modeling. The cell stream from a single voice source in a session can be characterized by arrivals at its peak rate during talkspurt and no arrivals during silence. We assume that successive taikspurt and silence periods form a two-state Markov process [3,17], and that talkspurt and silence periods are exponentially distributed with means pi~ I and

P01

on

P~o A. Single source ntP01

(nl- 1) P01

Plo

2plo B. Superposition of n~ sources

Fig. 4. Voice source modeling.

P01

nlplo

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

91

pol ~, respectively. It can be easily verified that the superposition of n~ such independent Markov processes is an n I + 1-state Markov process having the state transition diagram depicted in Fig. 4. Letting w~(t) denote the bandwidth of a single voice source, the equilibrium distribution of states (bandwidth) w~ can be written as Pr(w~ = i A l ) = ~{ E! 1 -e I

i=1 i=0

(8)

where I

Pol e =

Pm +Plo

Al = --

~9

Al

where AI is the peak bit-rate during talkspurts and i~ is the average bit-rate. The corresponding transform of thi~ probability, mass function is W , ( s ) = E [ e -~s] = I - e, + e, e --Al~ .

Laplace (I0)

The video source model that we use here is based on the one in [18]. Specifically, the bit-rate of a source is quantized into N + 1 levels and transitions between levels are assumed to occur with an exponential transition rate which depends upon the current level. This video source model can be obtained from a continuous bit-rate source by sampling at random Poisson time points wit!~ appropriate rates and quantizing the state at these time points. Assuming a peak bit-rate A2 bits/s, the video source is then modeled by an N + 1-state Markov chain whose steady state distribution is binomial and given by "A2 "-- (N)E/~)(1 - E2) N-i -----l~')

Pr(w 2

for

i=O,...,N

(11)

where A2 E

~"

~

(12)

.

A2 The corresponding

Laplace transform is

We(s) = E[e -ws ] = (1 - e 2 + ¢2 e-^"s/N) N.

(13)

It is known that data traffic is generally bursty and requires variable bandwidth [25]. We model data traffic in the same way as voice, viz., by an o n - o f f bursty source which is characterized by Pr(w3 = iA3)

___ [ e

3

1 -e

i= i 3

i =0

(14)

where B

(15) and the associated W3(s)

-"

Laplace transform is E[e -ws ] = 1 - e 3 + e 3 e -a3s.

(16)

Combining (10), (13) and (16), we find that the Laplace transform of a merged traffic stream of nl voice, n z video and n 3 data connections 2 on a link, is

W(s)

--[Wl(s)ln'[w2(s)]n2[w3(s)]

n3.

2 Data traffic can be modeled by (14)-(16) with a heterogeneous mix of e3i, i = 1..... n.

(17)

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

92

From the first and second derivatives of (17), we find that the mean and variance of the traffic stream are

(18)

E[w] =nlelAI + n2e2;t2 + n3e3A 3 and Var[w] = n~e~A~(1-e~) + n2N-=e2A~(1 - e 2 )

+/13/~3~t2(I --E3),

(19)

respectively. For traffic sources such as those characterized (10), (13), (16) and (17), as well as for any other sources of greater or lesser complexity, it is nccessary to have the means for rapid calculation of link weights in order to implement the call layer, including call routing. This can be done using the saddlepoint approximation [7,27], the sole requiremen, being that the probability laws characterizing the traffic models be Laplace transformable. To proceed, we note from the definition of link cost in (1) that D -- Pr( w > C) = fc=p(w) dw

1 fd+j® = 2rrj~d_j®s-t(l--W(s)) e ~c ds f o r d > O

(20)

where .~'{. } is the Laplace transform operator and j = ~/- 1. If we choose d < O, but greater than the real part of all singularities of W(s), (20) simplihes to 1

D=

fd+J=s-tW(s)e sc ds

2wj d-j=

for d < 0.

(21)

It is kpown from [7] that the integrand s-~W(s) e sc has a single minimum so in s < 0 along the real axis when C > E[~v]. so is called the saddlepoint since s-~W(s) e sc also reaches its maximum at so along the im,-ginary axis. As in [7], we define a phase function ~(s) by e '~'~ ~ s - l W ( s ) e sc

(22)

and expand it around the saddlepoint so to obtain

• (s) = ~ ( s o ) + ~q~"( So)( s - s0) 2 + o( s - s0) 2.

(23)

The first derivative does not appear on the right since so is an extremum of d~(s). If we neglect the higher order terms in (23), we then have s - I W ( s ) e ~c ~ e ~~,o+ ',"~,,x~-~o~' = sg IW(So) e soc e ½~'(~°x~-~.)'.

(24)

Substituting (24) into (21) and integrating along the path d -So, we obtain the approximation

W(so) O =

so~/2~"(so) e~"c

(25)

where so is the unique solution of d

• '(s)=~s[lnW(s)l+c-s-'=o

fors
(26)

Since qB(s) possesses a single minimum s o in s < 0 along the real axis, Newton's method is expedient and very effective in solving (26). We have performed several sample calculatiolls to link costs and found taat at most 15 iterations are needed to converge to a precision of 10-~2.

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

93

10 3

10 o

....................................................................................:: ..........................................:....................................... :

10-3

A

10-6

.... ~ ' " i ~ ....

c

i

i

_

.9 1o-9 10-12

........................................ "......B : C h e m o f f . b o u ~ d

l-skewed

exponent~al.rate.Poisson

............

C: Exact model (solid line) D: Saddlepoint (dashed line) E: Gaussian lOqS ......................................................................................................................................................................

lOqS 0.65

0.7

0.75

0.8

0.85

Link Utilization Fig. 5. C o m p a r i s o n o f link cost calculations.

The saddlepoint approximation for computing D k is simple and accurate. To illustrate its efficiency, we compare it with other approaches that can be used: the Chernoff bound based on the Poisson traffic approximation [I0], the Chernoff bound sharpened by the theory of large deviation and based on the skewed exponential rate Poisson tr~ffic approximation developed by Hui [10], and the Gaussian traffic approximation [10]. As an example, we consider a single link with slow motion video and virtual circuit data defined by (11) and (14), respectively. For the duration of a call, their respective average bit-rates are 5 Mbits/s and 620 Kbits/s, and their corresponding peak rates are 10 Mbits/s [18] and 1.544 Mbits/s (T1 carrier). The number of quantization levels for video is 11. Figure 5 exhibits the cell loss probability vs. link utilization for a mixture of 50% video and 50% data traffic on a 155 Mbits/s transmission link using our source models. The exact model (C) was obtained by detailed calculation of the call loss probability for all possible cases and serves as the reference for comparison purposes. The Gaussian approximation (E) over-estimates link resources and is, therefore, unacceptable for use as a link cost measurement because it may lead to a degradation in traffic service quality. Hui's (B) and the Poisson traffic (A) approaches are bounds and would sacrifice link utilization when used as a link cost since they under-estimate the link resources. The saddlepoint (D) approximation gives accurate results for existing traffic conditions and has a lower computational complexity than the Gaussian and Hui's methods, the former being the one that is the closest to both the exact model and the saddlepoint approximation. To evaluate the performance of the cell layer congestion control, we consider a general, time-slotted, ATM switch as illustrated in Fig. 6. The switching system is composed of a set of M-buffer input processors at the input ports, an internally nonblocking switching fabric with m-speedup (m >1 1), and a set of output processors for cell processing and transmission. For the case where m > 1, we assume infinite buffering at the output since we are interested in establishing theoretical performance limits. Cells that arrive at input processors are buffered and wait for transmission through the switching fabric. Once a tagged cell moves to the HOL, it joins the contention for successful transmission with cells at the head of the other input queues which are destinated for the same output port. The contention process at a tagged output k is mathematically represented by the relation

C [ , = ( C , _ m ) ++Ak

(27)

94

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN Output Processor

\

g Hi] iilLq

T Hil LqP]i'!Hfl ---m Input

ocessor

.jj j/j

i

HOL Contention

Fig. 6. ATM switching system with finite size buffers and output processors.

where Ck is the number of cells at the HOL of the input queues that are destined for output k at the beginning of a slot, A k consists of those cells which have moved to the HOL of the input queues during the slot and are destined for output k, and (.)+ denotes the larger of zero or its argument. The variable C~ then specifies the number of cells that have accumulated for output contention at the beginning of the next slot. Li [16] has shown that if the cell arrivals are Bernoulli with parameter 0 < A < 1 and the switch is arbitrarily large, then Ak is Poisson with mean ~ = 7. Under these conditions, (27) characterizes an M / D / m queueing system whose solution can be found in [23]. In the case of no speedup, viz., m = l, a closed-form solution for (27) is given in [14]. We note that the service time for an HOL cell is characterized by the Markov transition relationship (27) and a renewal probability distribution (which is the distribution of the states C k at the completion of the previous HOL cell being successfully transmitted). This distribution is termed the p h a s e distribution (PH) [22]. Neuts has shown that a continuous-time P H is asymptotically exponential [22, Theorem 2.3.1] if the Markov chain is irreducible, i.e., there exist K > 0 and 77 > 0 such that the probability distribution function F ( x ) --} 1 - K e -nx as x --} ~. Using an argument similar to the one in [22], it can be shown that a discrete-time P H is asymptotically geometric if (27) is irreducible. Hence, we take the HOL service time to be geometrically distributed with mean C--kk/~which can be evaluated by solving the Markov chain (27) for ~ and applying Little's result. We model cell arrivals at an input queue as a Markov modulated Bernoulli process (MMBP) based on the following observations: • The superposition of independent Markov chains that are used to characterize voice, data and video traffic is itself a Markov process. • The probability that the bandwidth of the combined traffic exceeds the capacity of a link or a switch is very small as a result of call layer control. This means that the bandwidth is rarely exceeded, and then only over very short time periods. Consequently, it is reasonable to assume that traffic flow is Markov.

Fig. 7. Transition diagram for cell layer queueing model.

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

95

• The cell stream is considered to be a Bernoulli process with a time-varying mean as a result of statistical multiplexing even though the traffic from a particular source is bursty. For example, suppose a compressed motion video source with an average rate of 5 M b i t s / s [18] shares a communication link of 155 Mbits/s capacity with a number of otht:l such sources. Then, on the average, a cell from each of these video sources occurs every 31 slots. Hence, a number of consecutive cells from a single video sources is unlikely to occur, and it is reasonable to assume that the cell stream is a Bernoulli process• We note that the mean of the Bernoulli process corresponds to the current traffic volume and can be described by a Markov process according to the previous observations. Next, we define the state of our queueing system by (u, v) in which u ~ {0, 1 , . . . , M } is the queue length at the input buffer in the current slot and v ~ {0, 1 , . . . , N} is the number of sources that are active when N connections are in place• The set of states {(i, 0),..., (i, N)} is called level i. If we assume that a cell which arrives after a slot boundary will be processed starting at the next slot, then input queueing at a switch can be modeled by a two-dimensional Markov chain whose transition diagram is depicted in Fig. 7. It is easy to see that the corresponding transition probability matrix has the form -Aoa

A~,o

Ao,2 Al,I A2,o Q __.

AI,2

A~,!

(28)

AM-l,! AM-

1,2

AM'° ] AM,!

where A i j , i = 0 , . . . , M; j = 0, 1, 2 are N × N submatrices that specify the state transition probabilities between two levels. It is known from the classical ergodic theorem of Markov chains that, if Q is positive recurrent, there exists a vector x & [Xo, Xo,2,..., X0,N,..., XM, N] which is a probability vector and also the left invariant eigenvector of Q, viz.,

xQ = x

(29)

xe = 1

(30)

and

where e is the column unit vector. One may note that (28) represents a finite-state queueing system with state-dependent arrivals and cannot be solved by the iterative techniques of the matrix geometric method [22]. We present here, however, an analytical solution for this particular queueing system that is specified by the following theorem• Theorem. If the Markov chain which is characterized by Q in (28) is positive recurrent and x i ~[xia Xi,2,...,XI,N], i =0, 1 , . . . , M , then (a) xi+ 1 = xiRi for i = 0 , . . . , M - 1, ( b ) the matrices R i, i = 1 , . . . , M are specified recursively by

R M = -AM,o(AMa - I ) - l , R i -- Ai.o( I - Ai. i - Ri+ 1ai,2) - !

for i ffi O, . . . , M - 1

where I is the identity matrix, and (c) Xo is the positive left invariant eigenvector of Ao, l + RIAo,2 normalized by xo( l + R l + R I R 2 + "'" + R I R 2 "'" R M ) e = 1.

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

96

Proof. The proof of this theorem is given in Appendix B. We now present some numerical results to illustrate: (1) the use of the above theorem to evaluate switch performance over a wide spectrum of traffic sources, and (2) the performance of the congestion control protocol that we have developed here. We consider a single link since routing problem is beyond the scope of this paper. In all cases, we assume that time is expressed in slots where a slot is a cell's transmission time on a link and that the term "input traffic rate" refers to the normalized average input traffic rate in units of cells/slot over the range [0,1]. Further, we assume a 256 x 256 switch with input buffer size M = 256, speedup of m = 1, 2 and 4, and average burst length of 64 and 128 cells. In Figs. 8 and 9, we show switch cell loss probability and average cell delay, respectively, vs. input traffic rate for m ffi 1 and five different input traffic types. We remark that type C is a superposition of 10 type A sources, while type D is a superposition of ~0 such sources. In both figures, the performance degradation in cell loss and delay that results when a single, two-level source dominates the switch inputs is dramatic as shown by the curves marked A in the two figures. At the other extreme, we have the results for type D traffic where it is clear that cell loss and delay performance of a statistically multiplexed traffic stream very closely approximates that of a Bernoulli arrival process. In any case, curves A and E in these two figures bound switch cell loss and delay performance. The results in Figs. 10 and 11, and 12 and 13, deal with the same aspects of perfarmanee as those in Figs. 8 and 9, respectively, but the former are for a speedup of m - 2 and the latter for rn = 4. The resulting performance improvement is significant and merits further attention. In particular, it should be noted that switches to achieve these speedups can be realized via either dilated switch fabrics [4] or bit-parallel switch architectures [13,19]. For a speedup of m = 4, Figs. 12 and 13 show that switch performance degradation is essentially insignificant, even at high input traffic rates, as traffic characteristics range across the spectrum from the "worst case" (type A traffic) to the "best case" (type C, Bernoulli traffic). Clearly, the use of speedup can serve to simplify the congestion control problem. Finally, it should be noted that Figs. 8-11 include simulation results which match analytical results very well. The total simulation run for each data point was 50,000 slots following a 5,000 slot "warm up"

10 6

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Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

97

100 / 90 ......... AL 2.le.v.e..l. correlated ~put.iC! 28..a.vg, ee!! b.~.s.0..i B : 2 ~vel correiated input ~64 avg. cell burst) ! 8o

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Fig. 9. Average cell delay for a switch with 256 cell buffers and no speedup vs. average input traffic rate for various bursty inputs.

interval. Only cell loss simulations down to a value of 10 - 4 for the loss probability were possible due to limited computing facilities and run time constraints. To illustrate the performance of our congestion control protocol, we consider a case with three classes of traffic where the cell loss requirements are R~ - 10 - 3 , R 2 - 10 - 6 and R 3 = 10 - 8 , respectively. We again consider a 256 × 256 switch with input buffer size M - 256, but no speedup. We assume that the

106

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Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

98

100

-r

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......................

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Input Traffic Rate Fig. I I. Average cell delay for a switch with 256 cell buffers and speedup of two vs. average input traffic rate for various bursty inputs.

three traffic classes are independent, but have the same characteristics. Namely, at one extreme, we take each to be a single 2-level, correlated input source with an average burst length of 128 cells and a common input traffic rate_~= A~= A-~= A cells/slot; and at the other extreme, we assume each is a Bernoulli source with rate A cells/slot. These extremes serve to bound the congestion control problem.

100 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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Input Traffic Rate Fig. 12. Cell loss probability for a switch with 256 cell buffers ~ d , inputs.

,,edup of four vs. average input traffic rate for various bursty

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

99

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Input Traffic Rate Fig. 13. Average cell delay for a switch with 256 cell buffers and speedup of four vs. average input traffic rate for various bursty inputs.

The results are s h o w n in Fig. 14. For the correlated traffic case, the total input traffic rate must be limited to no more than 0.434 cells/slot in order to satisfy all three cell loss requirements in the absence of congestion control. This limit is dictated, as one might expect, by the strictest requirement R 3 -- 10 -8, and can be implemented by throttling the sources such that 3A < 0.434. W h e n congestion control is

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100

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

applied, the total input rate limit increases to 0.477 cells/slot, a 10% improvement. Here also, throttling of the sources to prevent their exceeding this limit is needed. The limit itself can, of course, be increased by using larger buffers. But, at this limit, the call layer will not admit any new connections. For the case of Bernoulli arrivals, the improvement in the total input rate limit by using congestion control is not as dramatic as above, a result that is clear a priori from the slope of the cell loss vs. input traffic rate curves in Fig. 14. Without congestion control, we have 37 < 0.578 cells/slot vs. 37 < 0.584 cells/slot with it, an improvement of slight over 1%. At this point, we are essentially at the theoretical throughput limit of 0.586 cells/slot for the switch. While we have not considered speedup here, we know that the associated cell loss performance will be bounded by the results in Figs. 10 and 12 for m = 2 and m = 4, respectively.

4. Summary and conclusions We have presented a congestion control protocol for connection-oriented services in B-ISDN that is based on the traffic viewpoint layered design approach and implemented by a two-layer scheme. At the call layer, traffic flow is optimally distributed in the network by routing connections using a shortest path routing algorithm. Call admission, along with call routing, prevents network capacity overflow and guarantees that GOS requirements are met. The link cost for connection routing is defined for real-time traffic conditions and is calculated by the saddlepoint approximation technique. With the control of multiple traffic types in mind, a buffer reservation scheme at the cell layer was formulated to provide fair allocation of switch resources with respect to GOS requirements when congestion cannot be avoided due to heavy traffic. Cells being switched for each traffic type share switch resources and experience a loss probability proportional to their GOS requirements. Numerical results show that our congestion control scheme can reduce network congestion and improve the switching throughput, while satisfying GOS requirements for each traffic type. Our performance analysis employs a discrete-time finite state queueing system with geometrically distributed service time and state-dependent Markov modulated Bernoulli process arrivals to model the switching process. We developed an analytical solution for this model by extending the matrix geometric techniques [22] to state-dependent queueing systems. In contrast to the iterative approach generally used in matrix geometric calculations [22] for finding Ri, we provide a recursive form. This greatly reduces the computational complexity and increases the range of queueing systems that can be treated.

Appendix A In this appendix, we derive the input buffer limits Ki for the cell layer congestion control in Section 2. Input queueing for the switch is modeled as a discrete-time, finite buffer queueing system; and in order to obtain a closed-form solution for the Ki, w e assume that arrivals have an independent Bernoulli distribution and the service time is geometrically distributed. To simplify the derivation, we consider three types of traffic, which have average bandwidths ~ , A'~2,~aa and maximum allowed cell loss probabilities R~ >~R 2 >I R3, respectively. The probabilities of cell arrival in a slot for each traffic type, denoted by am, a 2, a3, are then given by the average bandwidth divided by switch capacity C. We denote by {rri} the probability of having i cells in the buffer and let w = [~ro, ~rl,... , rrM] be the corresponding probability row vector where M is the number of buffers at the switch input. It is easy to see that the transition probability n,atrix for this queueing system is q,.t

Q~

ql.2 ' . .

"..

(A.1)

qM- l,M- l

q M - l,M

qM.M- l

qM.M

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

101

W h e n the system is in steady state, we have the equilibrium relationship • r = 7rQ and ~r is normalized by ~re = 1 where e is the column unit vector. The steady state probabilities are then specified by q k - l,k

~rk=~rk_l~

qk,k- 1

(

where ~ro=

k

=TroI" I

M

1+

,

fork=l,...,M

(A.2)

i = 1 qi,i- 1

q i - l,i

)l

~ "=

q i - 1,i

.

(A.3)

qi,i-I

F r o m the definition of the K i, i = 1, 2, 3, and the cell layer control scheme d~picted in Fig. 4, we have q o J = a l 4- a 2 4- 0~3,

qo,o = 1 - qo,l, ql,o -" ~ ( 1

- - a I -- a 2 - - a 3 ) ,

ql,2 = (1 - / 3 ) ( a 1 4- U 2 4- a q 3 ) , ql,i - 1 - qt,o - ql,2,

qKl,Kl-i = f l ( 1 -- a 2 - - a 3 ) , qrl~~:~ + 1 = (1 - fl)( a 2 "[- a 3 ) , qrl,r., = 1 - qKl,Kl - 1 -- qK~,Ki + I,

qK3,K 3-1 "" ~ ,

qr3,r3 = 1 - qr3,r3-a

(A.4)

w h e r e fl is the probability that an H O L cell is successfully transmitted in a slot. Now let us define t~3(1 - - ~ ) P3 -" ~ ( 1 -- a 3 ) ' (a2 +~3)(1

--fl)

/ ~ ( 1 -- a 2 -- a3)

P2 --

'

o3=1, K 3 - K 2- 1

0"2 "--

E

P~ "[" P3/(3-K2( 1 -- a3)0" 3

i=0

1

-

p K3-K2

=

"Jr"p3K3-K2~30"3, 1 --P3

r2-r, - 1 io=

( 1 - a2 1

-

u3

0-2 U3

1 - p2r2 -KI + pK2-Ki.y20-2 1 -P2

(A.5)

Z. Ren, J.S. Meditch / A two-layercongestion protocol for B-!SDN

102

where T3 = 1 - a 3 and Y2 = (1 - a 2 - - a 3 ) / ( 1 - a 3 ) . The real-time cell loss probabilities, denoted by B l, B 2 and B 3 for traffic type 1, 2 and 3, respectively, are then given by B3 =

'ggK3 = Tr M M

B2 -

(A.6)

y ' ,rri i =K 2

and BI=

M ~ 7ri.

(A.7)

.

i=K!

Substituting (A.2)-(A.4) into (A.6) and (A.7), and using the definitions in (A.5) produces M-K

B3 - "fl'K,P3

1 -

B2"-WK:

2

Y30"3 '

p~-r2 + p~-r:y3tr3)

i --P3

= ,rrglpK:- Ki~,20" 2 , 1 -

+

(A.3)

i-o-;

B, = =K,

Applying traffic enforcement (2) from Section 2, 0<

B!

Rt

B2

B3

R2

R3

. . . . .

<1

to (A.8), we find after algebraic simplification that R2 Ra

B2 Ba

1--pT-g2+ ( 1 - p3)(P3M-K2y3tr3)

R2

B2

paK2-K'YlO'2(l -- P2)

(A.9)

and

(A.IO)

"

Rearranging (A.9) and (A.10), we obtain K 3 - M, Ki = Ki+ l + in

Ri+ i

1

yi+l~+l(1--Pi+l) +

1 /In

Pi+l,

i - 1,2.

(A.11)

It is clear at this point that (A.11) can easily be extended to n (>t 4) traffic types. The parameter ~ in (A.4) corresponds to the HOL contention probability, which is a function of the average traffic rate. We can estimate//in real time via the relation /~--/~k--

k-l^ 1 k /3k-l + k/3k

(A.12)

where /3k is the reciprocal of the measured service time in slots in which the k th cell is successfully transmitted. Obviously, if we assume E[/3 k] =/3 and Var[flk] ffi tr, the estimator (A.12) of/3 is unbiased, i.e., E[/~] ffi//and Var[/~] ~ 0 as k ~ o0. Normally, flk is a function of time because traffic flow in the network varies with time. However, when k is large, the estimate/~ of/3 k becomes a constant instead of

Z. Ren, J.S. Meditch / A two-layer congestion protocol f o r B - I S D N

103

tracking the actual value of flk since f l J k -* O. To resolve this problem, we choose a modified estimator as in [26] where

./~=~k= bk_,a~k_, + bk

1 ~-~'kilk'

(A.13)

b k =abk_ 1 + 1

(A.14)

where 0 < a < 1 is a fading^parameter. The estimator defined in (A.13) and (A.14) is also unbiased and has the property that Var[/3] --, (1 - a)tr 2 > 0 as k --> oo. The last term in (A.13) is ilk~b, = fl,(1 - a)/(1 - - ¢ * ) - * i l k ( 1 -- a ) =/= 0 a s k --* 00, which means that a new sample fik always makes contribution to ft. In practice, the calculations indicated here would be conducted using massive parallel processors, and changes in the thresholds would only be updated in the presence of large traffic changes. It should be clear that only large changes in traffic volume will result in readjustment of the threshold. For example, the addition of a batch of 50 to 100 voice calls would seldom result in readjustment of the thresholds.

Appendix B In this appendix, we present a proof of the theorem in Section 3.1. The statement of the theorem is repeated here for convenience: Theorem. If the Markov chain Q in (28) is positive recurrent, then (a) the steady state probabilities are given by xi+ l = x i R i

foriffiO,...,M-1,

(B.1)

(b) the matrices Ri, i - 0 , . . . , M are computed recursively by R M -

--AM,o(

AM,i

--

I) -l,

(B.2)

R i = A i . o ( l - a i , i - R i + l , , | i , 2 ) -1

fori=O,...,M-

1

(B.3)

where I is the identity matrix, and (c) x o is the positive left invariant eigenvector of Ao. I + RoAo.2 normalized by

(B.4)

Xo(J + R o + R o R l + "'" + R o R l . . . R M _ l ) e f f i 1.

Proof. We let JpU~ i,j;u,v be the probability of transition from state (i, j) to (u, v) in I time steps and introduce the definition of a taboo probability from [22]. Given two states (i, j), (u, v) and a set H of states in a Markov chain, the taboo probability H Ji,j;u,v ptl~ is the conditional probability that given the chain starts in (i, j) at time O, it reaches (u, v) at time l without having visited the state set H. We denote by (3O

R ( k ) .'~ i - - j r ~" E i"D(I) i,j;i+k,r !=1

the transition probability from state (i, j) to (i + k, v) without going back to level i and define R i "~ •

[ R<'J1 i''jr

I

-

RI' ,

R~°~ ~- [i[ R It' t°)]J = l

fori=l,

"""'

M.

By conditioning on time and the state of the last visit to level i, given that there is such a visit, we have the relation M P.l-r l , j ; i + l , j -_ i pV) i + l , j ; i + l,j

I

+ y" ---.. -y" - ~v(,) i+ l,j;i,v i fD(l-r). i,v,i+ 1,j v=l r=l

for 1 >i 1.

(B.5)

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

104

We add these equations for I ranging from 1 to L and divide the resulting sums by L. As L ~ ~, the left-hand side tends to x i+ tj by virtue of the classical ergodic theorem for Markov chains. Since the sum ~'~'l°==1 i'iP(l)+1,j;i+ l,j is finite, the first term on the right hand side tends to zero. The second term after summation over l is M1L l M 1 L L-r zi+ l,j;i.u i'i,u;i+ l,j "-~ 4 i+ l.j;i,v E i"p(l) E TE EP(') D(,-,, = E EP(') i,u;i + l,j v=l

l=r r=l

r=:

v=l

i=1

M -') E

Xi,v

iRvj

as L

~

u=-I

since (1/L)~L__- I D(r) ,i+ i.;;i,, ~ xi,v and E L I ' ,-, p ,,~:, 0) + ~,j ~ i R ~ ) for 1 < v < M as L -~ oo. We then obtain xi+ l - xiR~ for i = 0 , . . . , M - 1. This completes the proof of (B.I). Next, we show that

R 2)

=

(B.6)

r(2) t[ i"J,v] ----RiRi+ i.

From the definition of the taboo probability, we have M

p(I) i'i,j;i+2,v

=

!

E

i p(r) i , j ; i + l , h i + l P i +(l l,-hr;)i + 2 , v

E

h=lr=l

•

Summing over l, we find that E

.~(2)__=

t "'JU

E

p(r) (I-r) i" i,j;i + l,h i'~ 1Pi+ I,h;i + 2,u

i=lh=lr=l

= ~.

.plr)

(!'1 i+ie~i+i,h:i+2,v

i i,j:~+l,h" l'=l

h--! r=l M

E i'ljh

=

i+ ,el:2

h=!

which yields (B.6). From the definition of Q in (28), we see that = i ~ i , j ; i + I.u D(')

i" i+l,j;i.v -- ( A i . 2 ) j v ,

.p,~).

=

and it follows that M

-- E [ p ( i - l ) #,J;l+ t,u -- h__lti i,j:i+i,h

.p(I).

i-

"

(Ai,l)hv

-!- p(t-I) i i,j;i+2,h

"

(Ai.~)t,~] fort> 1 .

Summing again over l from 1 to oo results in Ri =

Ai.o + RiAi,!

o(2)A + " i "~.i, ~.

= A i ' ° + RiAi,I + R i R i + l A i , 2 '

(B.7) (B.8)

which yields (B.3) after algebraic simplification. Setting i = M in (B.8) and noting that AM, 2 = 0 , we obtain (B.2). Finally, from the equilibrium relations x - x Q and xe = 1, we have Xo = x o A o , t +xtA2. o

= x o ( A o a + RoAo,2).

Z. Ren, J.S. Meditch / A two-layer congestion protocol for B-ISDN

105

Hence, x o is the positive left invariant eigenvector of Ao, ~ + RoAo,2 for which normalization yields M

~ x~ = x o +xoR o + ' ' '

+xoRoR

l ... R M_l

i=1

=xo( l + Ro + giving us (B.4).

"'" +RoRl

"'" R M _ I )

=1 []

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