Queueing Behavior under Flow Control at the Subscriber-to-Network Interface for High Speed Metropolitan Area Networks Izhak Rubin and K. David Lin Department of Electrical Engineering, Engineering IV,56- 125B University of Califomia, Los Angeles, Los Angeles, CA 90024

Abstract A Credit Manager Algorithm (CMA) is used to control the flow of packets between a subscriber station ( or a local area network ) and the access to a high-speed metropolitan area network. The parameters of the algorithm are used to regulate the mean throughput and the level of burstiness at the Subscriber-to-Network Interface (SNI). Such a flow control procedure has been recently considered by the BELLCORE'S Switched Multimegabit Data Service (SMDS). We study the queueing behavior induced by such a flow control scheme. We examine the increased queue size and message delays incurred at the subscriber's station buffer due to the controlled access mechanism, as a function of the underlying system parameters. We also investigate the statistics of the output process by characterizing it as evolving among idle, burst, and restricted modes. We present performance curves illustrating the behavior of the station queues and of the regulated output traffic streams. 1. Introduction For many communication networks, conventional end-to-end window flow control schemes can yield unsatisfactory delay-throughput performance trade-off and exhibit fairness problems. These problems are becoming more significant as the speed of transmission increases and the statistical variations of the mixed-services offered traffic grow. For high speed networks, it is important to have a different flow control mechanism to ensure acceptable performance. The concept of input rate control, so-called "leaky-bucket" method, was employed in [2][3]. The performance behavior of such a method is critically dependent upon the bursty nature of the offered traffic. In the IBM's PARIS[4] experimental packet switched network, an "input throttle" procedure has been adopted to regulate traffic. A queueing model is presented in [5] to analyze this scheme. The analysis presented in the latter paper assumes zero packet transmission times and the same token ( credit )

requirement for all packets, independent of their packet lengths. A general flow control scheme employing the Credit Manager Algorithm has been presented by Bellcore[ 11 to administrate the traffic at the subscriberto-network interface (SNI) for high speed Metropolitan Area Networks. Such an algorithm regulates the admission of traffic at the SNI, allocating different access parameters to different access classes. On the customer premises equipment (CPE) side, the credit manager algorithm protects the customers from being overflowed by the network; therefore, they need not allocate extra resources at the interfaces. On the other hand, network providers have to provide only the actual required resources to buffer the incoming traffic. The credit manager algorithm is implemented in the SMDS Interface Protocol ( SIP ) at layers SIP2 and SIP3 [l]. At both the CPE and the MSS ( MAN Switching System ) access points, the system has to transmit and accept data in accordance with the algorithm, as summerized in the following. For the transmission of a SIP packet, the system has to acquire sufficient credit, depending on the size of the packet, before it can be accommodated. A basic entity, of a prescribed length, called an Information Unit (IU)is defined. Each packet can contain multiple Ius. The system checks the number of Information Units contained in each level 3 Protocol Data Unit ( L3-PDU ), compares it with current available credit, and determines whether to let the packet pass through the SNI or delay it. The system then subtracts the required credit used for the packet from the total available credit, if the packet is accepted for transmission, and then it proceeds to transmit the packet across the SNI. The available credit is incremented by one every At seconds, noting that the maximum credit value allowed is equal to the upper limit Cma. Assume each Information Unit to contain p bits and the transmission rate of the link to be v bits per second. The average throughput across the SNI is set to correspond to the transmission of p bits every Af seconds. This rate is designated as the Sustained Access Rate (SAR). The maximum number of bits that can be transmitted when credit reaches C ,, defined as MAXBURST, can be expressed as 161 :

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containing L segments ) every K slots, so that SAR

L1

where x denotes the maximum integer not greater than x . Equahon (1.1) can be simplified to yield equation (1.3) when condition (1.2) is met :

x,

CnUlX
SAR

MAXBURST = C ,

(1.3)

Using this regulation algorithm, we observe that the transmission process at the SNI fluctuates between idle and busy periods. A busy period would last for up to MAXBURST seconds, with the average utilization of the SNI link equal to

E . In

2. A Queueing Model A discrete time queueing model is presented here to analyze the stochastic behavior of the system under the flow control mechanism implemented according to the Credit Manager Algorithm. We use this model to obtain the stochastic properties of the system size and queueing delay at the sender and derive the characteristics of the packet process passing across the SNI. This allows the system designer to asses the packet delays at the sender induced by the algorithm, as well as to determine the statistical characteristics of the regulated traffic process offered to the MAN. A set of recursive equations are presented to model the system's dynamic behavior. Simulations, using these equations, are then employed to obtain system performance results. Our system model is depicted in Figure 1. The sending side has to buffer its packets until the credit manager algorithm permits their transmissions. An infinite buffer capacity is assumed. The sending side could represent the CPE or the MSS. A slotted channel similar to the one used in a TDMA system is adopted ( figure 2 ). Time is segmented into fixed length slots; each slot is of duration equal to z [sec]. We define a segment to contain a bits and to designate the message entity whose transmission time is equal , v [bps] is the to a single slot, so that T = Swhere

v.

Credit ( in Ius ) available at the start of slot n.

c, =o, 1,

C', :

" '

,c,

Credit ( in IUS) after deduction, if any, at the start of slot n. c ', = 0, 1, . , c , Number of arriving packets during slot n and, as recorded at the end of slot n. ' '

A,

:

the following section, we

present a Qeueing model which we then use to derive the stochastic performance of such a credit manager algorithm.

K

The following variables are then defined. System size ( number of packets queued in : the sender's buffer ) at the start of slot n, n=l, 2 , 3 ... x,=o,1,2 ..'

c, : xp

L

=-

A,, = 0, 1,2, . .

s, :

v,

:

N, :

M,, :

Service requirement ( measured in slots ) of the packet at the head of the queue at the start of slot n. S, = 0, if X,, = O S, =o, 1,z, . . . ,L x C , ; Number of remaining slots required to serve the packet at the head of the queue at the start of slot n. V , = 0, if x, = 0 v,,= 0, i,2, . . . ,LXC,; Number of slots elapsed since the last credit increment at the start of slot n. N, = 0,1,2, . . . ,K-1 Transmission indicator function at the start of slot n, where we set M , = 1, when slot n is busy (i.e., used for a segment transmission) M, = 0, when slot n is not busy.

Denote I (fluflags) to be the indicate function such that it equals 1 when all its arguments are true, and 0 otherwise. A set of recursive equations are written to describe the operation of the CMA, as presented in the following. The variable S represents a random variable governed by the packet service time distribution. N,,=(N,-l+l)mod K

s,

(2.1)

=

transmission rate across the SNI. Slot n occupies the period ( ( n - l ) ~n, z ). We set a time frame to be equal to the time between credit increments ( A t ) . A time frame is thus selected to contain K slots, where we set At =Kz. Each packet is assumed to contain an integer multiple of segments. An Information Unit (IU) is set to consist of L segments, where L is a positive integer. On the average, the sender is thus permitted to transmit a single IU (

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(2.3)

Using these equations, we have developed a simulation program for studying the performance of such a system. In section 5 , we also present an analytical solution methodology, when considering single-segment packets. Results are presented in the following sections.

In figure 4, the effects of,C on the average system size are demonstrated. The time between credit increments is fixed to be Af = 20 slots, the IU is equal to L =5 segments, and the packet length is uniformly distributed between 1 segment and 10 segments. ,C is used as a parameter for each curve. In all cases, the same maximum throughput rate is achieved. However, the larger the C , value, the smaller the average system size ( when the system is stable; i.e., when the input rate is lower than the maximum throughput level). When ,C is large, the algorithm imposes less regulation on the arriving packets, causing reduction in packet queueing delays. As a result, the average system size at the interface is reduced. In Figure 5, we illustrate the effects of different IU lengths on the average system size. We set the time between credit increments to be K =20 slots, the maximum credit is C, = 10, and the packet length is uninformly distributed between 1 and 10 segments. The Information Unit length, L , is used as a parameter in this figure. The average maximum segment throughput rate thus increases as L is increased. We observe the stability condition for the CMA to be :

3. Performance Results for the Average System Size kE In this section we present our simulation results concerning the behavior of the average number of packets ( system size ) queued at the sender’s buffer as a function of the offered traffic load. Using Little’s formula these results can also be used to compute the mean packet queueing delay in the sender’s buffer. Figures 3, 4, and 5 demonstrate the effects of different parameters on the average system size. The arrival process is assumed to be Poisson. In figure 3, we fix the time between credit increments to be equal to 20 slots ( K =20 ), the Information Unit length to be 5 segments ( L = 5 ), and the maximum credit to be 10 ( C=, 10 ). The average system size is plotted versus the arrival rate, which is normalized to represent the average number of packet arrivals between credit increments. The packet length, s, is used as a parameter for each curve, and is uniformly distributed between the limits indicated on the curves. The results show that for short packets the maximum throughput of the system is close to l/Ar [packets/sec]. As the packet length grows, so that each packet requires more than 1 credit, the maximum packet throughput rate is reduced as expected. For example, consider the case where packet lengths are uniformly distributed from 1 to 15 segments ( S = 1,15 ). The average packet length is then equal to 8 segments requiring a credit of at least 2 (TLJs) for its transmission. As a consequence, the maximum throughput level is reduced to 0 3 A f [packets/sec].

[ S] c K

(3.2)

The left hand side of (3.1) represents the arriving credit demand rate measured in IUs/frame, which should be lower than the provided system credit generation rate of 1 IU/frame. Equation (3.2) ensures the offered segment load per frame, L E S , is lower than the frame maximum service rate of K [segments/frame].

[I

4. Output Traffic Characterization In this section, the sender’s departing processes across the SNI where the Credit Manager Algorithm is used, are characterized. We observe that at any given time the departing process could be in one of the three modes; idle, burst, or restricted modes. In idle mode, there are no packets queued in the sending buffer while a continuous transmission of packets takes place when the output process is in burst mode. In the restricted mode, the state of output process alternates between periods during which the sender waits for sufficient credit to accumulate and periods representing each subsequent single packet transmission. At the termination of a restricted mode, the process switches to idle mode. The output process state switches from burst mode to restricted mode when it runs out of credit. A complete definition of the three mode characterization of the output process is given as follows. We n-rh mode M o d e ( n ) = k , k E define the

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I

a

Idle, Burst, Restricted

, and Mode( n

)

starts at slot i

and ends at slot j if an only if : case (1) k =Idle : Mode ( n-1 ) = Burst or Restricted, Xi-, > 0, Xi = 0, X; = 0, and X;+l> 0. j is the smallest integer not less than i case (2) k =Burst :

.

M o d e ( n - l ) = I d l e , X i - l = O , X i >O,Mi-l=O,Mi =1, M; = 1, and M j + l = 0. j is the smallest integer not less than i

.

case (3) k =Restricted :

{

( a ) M o d e ( n - l ) = I d l e , X i - l = O , X i > O , M i =Oor

I

@ ) M o d e ( n-1 ) = B u r s t , M i - l = 1,Mi =O,Xi > 0 and X j > O,Xj+l = O j is the smallest integer not less than i . As defined in page 5 , Xi and Mi are the system size and transmission indicator at slot i, respectively. The traffic characteristics are different in each mode. While in idle mode, the system is idle; this continues until the arrival of a new packet. When in burst mode, the sender transmit across the SNI Under restricted mode, the credit manager algorithm regulates the sending side so that it has to wait a period of time ( whose duration depends upon the length of the current packet and the current available credit) before it can transmit a packet. Our recursive equations were used to compute the steadystate distribution representing the probability of occurrence of each mode and the average duration of stay in each mode. Figures 6 to 11 represent the results for three different sets of data. The occurrence probability and average duration for each mode are plotted versus the arrival rate; the latter is normalized to represent the average number of arrivals between credit increments. In figure 6 and 7, the time between credit increments is set to K = 20 slots, an IU contains L = 5 seg10, and the packet ments, the maximum credit is C=, length is uniformly distributed between 1 and 10 segments. In figure 6, we can see that the probability that the system is in idle mode sharply decreases as the input rate increases beyond a certain threshold level. The probability that the system is in burst mode increases when we first increase the the input rate. As we further continue to increase the input rate, the system runs out of credit more often and enters the restricted mode. As a result, the probability of being in burst mode decreases while the probability of being in restricted mode increases. In figure 7, the average duration of stay in burst mode is observed to be approximately constant. We note that when the input rate is small, the traffic rate is not sufficiently high to induce long stays in the burst mode; while as the the traffic intensity grows, there is not sufficient credit to sustain a long stay in the burst mode either.

In figures 8 and 9, we reduce C, to 2 while keeping other parameters the same values. In comparing figures 6 and 8, we note the effects of reducing C, to lead to the decrease of the probability of system residence in idle and burst modes and the increase of the probability of Occurrence of restricted mode. When comparing figures 7 and 9, we observe that the main significant difference occurs at the start of the restricted mode. As the input rate increases beyond a certain level, the system runs out of credit more often and the average duration in restricted mode is insensitive to the actual C maJr value. In figures 10 and 11, we set the packet length to be uniformly distributed between 1 and 5 segments and keep the other parameters as in figures 6 and 7; i.e., K = 20, L = 5 = , ,C 10. In comparison with figures 6 and 7, one observes that the probability of the system being in idle mode is now noticeably reduced at a higher input rate. Similarly, the probability of restricted mode occurrence increases at a higher higher traffic rate. These effects are caused by the reduced need of credit required for the accommodation of the shorter packets. The average duration of stay in burst mode in reduced due to the shorter packet lengths. However, the average duration of stay in idle mode stays about the same, due to the memoryless characteristic of the Poisson arrival process.

5. Special Case : ATM Messaging Structure Under an Asynchronous Transfer Mode ( ATM ) operation, each packet consists of a single segment, also denoted as a cell. Each packet can thus be transmitted across the SNI when at least a single segment credit is available. The process

{

X,, ,C, ,N,, , n 21

I

is now a discrete-

time Markov Chain, whose evolution is governed by the following recursive relationships. The number of departures in slot n, D,, ,is given by : D , = I ( X , >O,C, > O )

(5.1)

The variables X , , C, ,N, ,and A,, are as defined in section 2. The constants C, and K represent the maximum credit and the time between credit increments ( in terms of slots ), respectively. We have : X , , + l = X , + A , -D,,

(5.2)

C,+l=min[ c , - D , , + I ( N , = K - ~ ) , c , , ]

(5.3)

N,,+l=(Nn+l)mod K

(5.4)

The transition probability function for this Markov Chain can be obtained as follows. Equation (5.5) states the relationship between the joint probabilities and

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certain conditional probabilities, when considering a randomly observed time slot n . P(X,=i,C,=j,N,=k)=P(X.=i,C,=j

IN,=k)P(N,=k)

TO compute =1 K P ( X , = i ,c.= j 1 N.= k )

(5.5)

Define u, = P ( A . = m ) , m =O, 1 , 2 , . . . . For i 20, the transition probability function is then expressed by the following equations. When ,c z I ,we have : Case (A) K - 1 s =k z 0 : (a) For c"+~ = j = 0, P ( X , + , = i , Cn+l=O,Nn+l=k) = ~ ~ P ( X , = i - m , C , = O , N . = k - l ) m=O

+ x u m P(X.=

i-m+l, C.= l,N.= k-1 ) z

R( i , j

Ik

)

I

, we need to solve the fol-

lowing set of equilibrium equations, obtained directly from Equations (5.6)-(5.12). For i ;? 0, we have : I

I

%,O&

=

am xi-m,O,k-l

am Xi-m+l,l.k-l

m=O

md)

(5.15)

j=O, K-lZk>O 1

%j,k =ai

n0J.k-1

+

am n i - m + l j + l , k - l md)

C ,,

(5.6)

n=O

(b) For c,,

{

> j > 0, K - 1 2 k > 0

(5.16)

j z 0,

P (Xatl= i, C.,l=j,N.+1= k )=ail' (X.= 0, C.=j,N.= k-1 )

+ ~u,P(X.=i-m+l,C,=j+I,N.=k-l).

(5.7)

m=O

= c,

(c) For j

P(X.+l=i,C.+l=C,fln+l=k

(5.8)

)=u,P(X,=O,C.=C,,~"=k-l).

j=l,k=O

(5.19)

C,>j>l,k=O

(5.20)

Case (B) N . , ~= k = o : (a) For c.+~ = j = 0, P ( Xn+l= i, c.+1=0, Nn+l=0 ) = 0.

(b) For

(5.9)

=j = I ,

P ( Xn+l=i , Cn+l=~ , I V . , ~ =0 ) = Ccq,,f(X.= i-m, C.=O,N.= K-1 ) *=O

+ Zu,P

(X.= i-m+l,c.= 1,N.= K-l ) .

+

(5.10)

C a m Ri-m+1,CmUp-l

Cmax=j* k = O . (5.21)

m=O

m a

(c) For c,,

This set of linear equations can be numerically solved, under the boundary condition :

> j z 1,

P ( Xntl= i , Cn+l=j , N.+,= 0 ) = u i P (X.= 0, C.= j - 1 , N,,= K-1 )

00

cmu

C Cx( i , j I + Cu,f(X.=i-m+l,C.=j,N.=K-l).

(5.11)

n=O

(d) For j = c,,, P (Xntl= i,C.+l=C,,N.,l=O)=uiP

+ UiP (X.= + C u,P

0, c.= C--L

(X.=O,C,=C,,,N.=K-l)

N,= K - 1 )

m=O

The steady-state conditional probabilities, when they exist, are given by : n i j k EX(

(5.13)

i, j I k ) = "+ l i m f ( X . = i , C . = j , IN,, = k ) ,

The joint steady-state distribution of (xn, C. given by :

)

is then

for each k , 0 Ik I K - 1

. (5.22)

A computer program based on Gauss-Seidel method has been used to solve these equations, under a properly truncated state space. To illustrate performance, a geomemc-batch arrival process is considered. Packets arrive in batches in accordance with a geometric point process; each batch consists of B packets, B > 0. Then we have :

(5.12)

(X.= i-m+l, C.= C,,, N.= K-1 ) .

k ) = 1,

i=O j=O

U,,,=

I

p , I-p, 0,

h=C mma,=Bp

m=B m=O

(5.23)

otherwise

,

B>O,O
(5.24)

Figure 12 illustrates the performance in terms of the average system size for different batch sizes ( B ) and

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maximum credit values (C,), as the input rate (A) increases. We fix the time between credit increments to be equal to 5 slots (K = 5 ) . One can observe that for a higher batch size, the credit manager algorithm induces a higher delay at the interface. Due to the bursty nature of the batch arrival process for high batch size levels, it takes longer time before sufficient credit accumulates for the transmission of all the packets in a batch. For a , value, the algorithm runs out of credit more lower C often, thus leading to higher queue levels.

6. Conclusions In this paper, we present a set of recursive equations to model the operation of the Credit Manager Algorithm. This algorithm is used to regulate the flow of packets at the subscriber-to-network interface (SNI). Simulation results are presented for the average system size at the SNI, representing the packet delays at the sender's buffer. These results can be used to determine system and algorithm parameters required to ensure an acceptable packet queueing delay at the sender's buffer. The regulated output traffic across the SNI is statistically characterized by categorizing its states into three modes; idle, burst, and restricted modes. Such a statistical model for the output process can then be used to investigate the applications of the regulated traffic stream to a metropolitan area network. We also present a model for the queueing behavior of an ATM type system, where single segment packets ( cells ) are served. A numerical method is used to solve for the underlying system state probabilities.

sending side

or the s y s t e m Depart

i o the receiver

Arrivals Server

Figure I S y s t e m Model for t h e C r e d i t M a n a g e r R l g o r i t h m

v/

Frame p o s i t i o n index

Figure 2 The S t r u c t u r e o f t h e S l o t t e d C h a n n e l

Figure 3 Compmnron of Packet Lengths

7. References Generic System Requirements in Supports of Switched Multi-megabit Data Service; Bellcore TA-TSY-000772, Issue 2, March 1989 Jonathan S . Turner, New Directions in Communications ( or Which Way to the Information Age ?), IEEE Communications Magazine, Oct. 1986, Vol 24, No. 10. Jonathan S . Tumer, The Challenge of Multipoint Communication, 5th ITC Seminar, Lake Como, Italy, May 1987. Israel Cidon and Inder Gopal, PARIS: In Approach to Intergrated High Speed Private Networks, to be published. Moshe Sidi, Wen-Zu Liu, Israel Cidon, and Inder Gopal, Congestion Control Through Input Rate Regulation, Proc. IEEE GLOBECOM'89 Conference, Dallas, Texas, Nov. 1989. Celina S. Albanese, Interdependencies Among the SMDS Credit Manager Parameters, Bellcore, Technical Report, 1989.

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Figure 4 Compmron olCmia Valucs

I

Figure 6 Steady-Slate Distnhuiion 01 1i:ich M d e

Figure 5 Compdnronof IU Lengths 2JO -

V 200-

I I I I I

0

I

150-

'

,

ResmnedMade

I I

I I00

I

-

I

9

I

d D

so

-

.

..'

1 I I

8=.+. .' 0

/

"-p

0

0.0

02

'

Idle Made

d 1

4BuntMode I

I

I

I

0.4

06

0.8

I .o

Normalized Input R r c ( no of arrivals pcr time frame )

Simulation Rcrullr ( K.20. Cmu=lO. S=I-IO)

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2SO

-

9

200-

I I I I

I

.

Iso-

I RcsuincdMode Resvlned Mode

I

-

I

P

I 100

I

I

-

I

0.

I

0.

I

I

6

I

t Q,

so

-

.

@

0 00

C

?

D.

1

..

/ /

i t - .. e-.

A'

w c g

t

C

C

d

-0 ldlc Mode

QBuniMDdc

I

I

I

I

I

0 2

04

06

0.8

I.o

01

4C.2.8. 0389 ~

Average System Sile As a Function of Arrwal Rare ( Brrch arrivals are assumed. )

&mai=3.8=8

I

d

I

9 ~ x = I O . B = 8

p’! : 4 I

!

P f

00

0.2 0.4 06 0.8 Normalized Input Rate ( no of amvals pertime framc )

to

Fig. 12 Numcncal Results ( K=5 IS a s s ” )

Acknowledgement This work was supported by NSF Grant NCR8914690, Pacific Bell and MICRO Grant 90-135, US WEST contract D 890701, and a UCLA Academic Senate Grant.

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