Study of communication interfaces for multiple bus networks

177

A Study of Communication Interfaces for Multiple Bus Networks I. C H L A M T A C

a n d O. G A N Z

Technion, Israel Institute of Technology, Computer Science Department, Haifa 32 000, Israel

A model is presented which reflects communication interface design considerations present in emerging multiple bus communication. The model captures various levels of functional duplication in the receive/transmit mechanisms of the communication interface units. It is shown that the level of duplication significantly effects performance, especially for s y s terns with small number of communication units.

Keywords: Distributed systems, high availability interconnection, communication interface, performance modeling,

Imrich Chlamtac is a member of the faculty in the computer science department at Technion, the Israel Institute of Technology. Previously, he was w i t h the computer science department at the University of Minnesota and with Digital Equipment Corporation. He

received his BS and MSc from Tel i ~

Aviv University in 1977 and his PhD in computer science from the University of Minnesota in 1979. His interests include computer communication,

local area networks, and simulation,

He is coauthor of Local Networks, published by Lexington

Books. Ora Ganz was born in Bucarest, Rumania on August 20, 1957. She received the B.Sc. and M.Sc. degrees in

Computer Science from Technion-

Israel Institute of Technology, Haifa, in 1980 and 1983 respectively. She is presently working towards the Ph.D. degree in Computer Science Department at Technion. She has been engaged in research in the fields of corn-

purer communication, satellite cornmunication and queueing systems.

North-Holland Computer Networks and ISDN Systems 9 (1985) 177-189

1. Introduction Distributed systems configured around a multiple bus interconnection are gaining increasing attention [3-9]. The multiple bus configuration presents an attractive way for obtaining increased availability, easy extendability and incremental system growth. In this configuration the issue of bus communication interface unit (BIU) design must be considered in terms of cost and performance. Clearly the replication of communication software and hardware can improve the multiple bus system performance but at increased cost. In existing systems the whole-spectrum of functional duplication has been covered in the BIU various designs [3,5,9,11]. Several of these systems support more than one level of duplication as required per application. In this paper we investigate the implications of the various BIU design altermatives on system performance by proposing a model that permits us to evaluate the performance benefits gained by the additional BIU functionality. Since in practice most multiple bus systems use two buses (e.g. with nyperchannel networks 85 % of multiple bus configurations use two buses) we concentrate our attention on double bus interconnection systems. In thiS configuration, the BIU can be connected to both buses without duplication; i.e., it can receive or transmit on one of them [5,11]. Partial duplication enables the reception and transmission on different buses simultaneously [3,5,11]. Lastly, with full duplication the BIU permits simultaneous reception on both buses [3,5]. Increased parallelism will improve performance but requires partial or full duplication of the serial bit interface between the physical and data link controls and duplication of data encoding decoding and error detection functions. Additional hardware mechanisms e n abling parallel processing, additional buffers and parallel signal conductors, e.g. for transmit/receive signals, carrier presence or collision detection may also be required. Existing analytic models of multiple bus systerns assume full BIU duplication [7,8] and infinite

0376-5075/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

L Chlamta~. O. Ganz /Interfaces for Multiple Bus Networks

178

population [4,7,8]. The model presented here captures the various duplication levels while assuming finite user population. Thus effects such as throughput loss due to simultaneous message arrivals to a partially duplicated BIU which can receive on one bus at a time only, can be accounted for. The effects of partial BIU on system performance are most pronounced for small systems and capturing them in the model is thus particularly important since in practice the number of BIU's is not necessarily large. For instance, among installed Hyperchannelnetworks 95% containless than eight BIU's. The presented performance models deal with both synchronous and asynchronous communication systems. They can be used in deciding on the level of BIU duplication needed given system requirements and traffic loads and can thus be useful to the designer and the users of such systems.

is done in a way intended to reduce conflicts associated with the communications interface, BIU, realization. Specifically the BIU can be realized at various levels of duplication, of the transmit and receive mechanism: Case A : A node can transmit or receive on one of the buses at a given time (minimal duplication). Case B: A node can transmit on one of the buses and receive on the other simultaneously. Case C: A node can transmit on one of the buses and receive on the other or it may be receiving on both buses simultaneously (full duplication). Consequently, the level of BIU duplication places constraints on the selection of messages for successful transmission since depending on the case incorrect choice of messages may cause conflicts. We define a destination conflict to occur if more than one user attempt to access the same destination node (possible in cases A and B). A source conflict is defined as the situation in which the same node is selected as a destination node as well as the source node for independent transmission (possible in case A). With these assumptions, at a given time, each node can be in one of three different states: 1) the node generates messages, i.e. is executing locally, or is receiving a message - such node is defined as active; 2) the node is transmitting message; 3) the node has a message queued for transmission. We make the following assumptions regarding the system operation: 1) Message arrivals to the active node are assumed to be generated from a Poisson process with average arrival rate )~, s.t. the node message arrival is a Bernoulli process with rate 1 - e -x. 2) A uniform reference model is assumed; this

2. The Model This study considers a distributed system (see Fig. 1), in which each processor has local memory and local I / O interfaces. Execution of local code is out of the local memory and I / O is done through the local interfaces so that no bus cycles are used. When a bus transmission is required local processing is suspended until the request is filled. Measurements have shown that this basic model is valid for both distributed multiprocessor systems and local area networks [3,4,5,10]. We consider l) synchronous systems [1,2,9,10], i.e., the cycles on the two buses are synchronized and 2) asynchronous systems [3,4,6,9] where transmission times on the two buses are not correlated. In both systems the selection of messages for transmission

Nnodes

bus adapters

>

<

II

I1

!1

>

bses

Fig. 1. Block Diagram of a System with N processors and two Buses.

L Chlamtac, O, Ganz / Interfaces for Multiple Bus Networks

implies that a bus access request from every node is directed to any other node with equal probability. Thus, the access rate from node i to node j ( V i , j, i --/:j) is X / ( N - 1). We consider systems with two modes of transmis-

ynchron,,u

The average number of queued messages will be: L u = W . X* (4) To derive X* and L we must distinguish between:" a) Synchronous system. The rate X* is given by:

asynchro o

In the synchronous mode messages are of constant size with constant service time 1/~t. Up to 2 messages can be randomly selected for service at the beginning of each time slot in such a way that no conflict will occur according to the BIU realization, In the asynchronous system messages length are an independent exponentially distributed random variable with mean 1/~t. When a bus goes idle the next node to use the bus is selected at random anong nodes wishing to transmit in such a way that no conflict will occur according to the BIU realization,

X* =

We assume that the "message selection for transmission" process is done in negligible time as compared to the message transmission time. This assumption is viable since a number of protocols can provide this service in total delay equal approximately to the signal propagation delay [3,15,16,17]. For geographically restricted systems, e.g. multi-processor systems or backend storage networks this delay will be on the order of microseconds or nanoseconds. Messages generated at the nodes join queues and must be granted bus access rights to be transmitred. In the synchronous system up to two access rights are given at the beginning of a time slot while in the asynchronous system a permission is given whenever there is an idle bus and a queued message whose transmission will cause no conflict, The permit is returned upon the completion of a transmission. To derive the performance measures we proceed as follows. We define pi to be the probability that at the end of a service interval there are i messages in system, Define X* to be the rate at which messages cycle through the queueing network. Define L to be the average number of messages in the system, Applying Little's result we obtain the average access time D D = L/X* (2) and the average queueing delay W by substracting from D the service time 1//z

X*= (N-L).X and

W

L X*

1 /~"

179

(3)

N-

ipi (1 - e -

(5)

obtained by multiplying the expected number of active nodes at the end of a service interval by the probability of message generation during the time interval. The average number of messages L will be: L = N - 1- e x( ~ \ N-

N ~ JPi j = 0

(6)

The calculation of L is given m Appendix 1. b) For the asynchronous system we have (7)

N

L = ~ iPi. i=0

3. Exact Models 3.1. Synchronous System

To model the behaviour of the system for cases A and B under the assumptions given in the previous section, we observe the state of the systerns at the end of service time interval to obtain a discrete time Markov chain. The state of the Markov chain is defined by the N-tuple ( S 1. . . . . S u ) (8) where S~ state of processor i. S~ can take the values 0 (active), and j (queued for processor j at the end of a service time interval). When a processor can transmit on one of the buses and receive on the other simultaneously, i.e., case B, using the theory of " l u m p a b l e " Markov chains [14] we may lump equivalent states to obrain a substantially smaller number of states. The state definition of the exact lumped chain is given by (ql . . . . . qN)

(9)

I. Chlamtac, O. Ganz / Interfacesfor Multiple Bus Networks

180

where qi is the n u m b e r of messages waiting for a p r o c e s s o r at the end of a service time interval a r r a n g e d in decreasing order, ~'Ei,j be the p r o b a b i l i t y of exactly i packets originating from a b i n o m i a l process with rate l - e - x in a service time interval given that at the begin-

T h e level of a state l is defined as the n u m b e r of packets in the system. The n u m b e r of states at level 1 is: / S ( l ) = ~ Pi ( l ). i=o

ning of the service time interval there are j p a c k e t s in system ( N - j active processors).

T h e n u m b e r of states thus becomes

Let

qT"i,j~-

J)

i--

(1-e-X)'(e-X)

N J-'.

S=

(10)

W i t h ~rcj we can find the set of state p r o b a b i l i ties p~'s, a n d then derive the required p e r f o r m a n c e measures. A n e x a m p l e of the equations of the exact chain for N = 4 is given in T a b l e 1. It is obvious even f r o m this e x a m p l e that the n u m b e r of states increases r a p i d l y with system size a n d renders the m o d e l impractical. To rigorously derive the n u m b e r of states of the exact l u m p e d

=

i

Pi(J)

-1

(11)

To o b t a i n an exact m o d e l for case A the state definition m u s t a d d i t i o n a l l y r e c o r d the message source. Exact solution to case A is then achieved as described for case B. Clearly, so for case A the n u m b e r of states is also explosive.

3.2. Asynchronous Model

chain for case B, we define p k ( n ) the n u m b e r of u n o r d e r e d p a r t i t i o n s of n into k parts with k a n d

Using the a s s u m p t i o n s of the previous section

n integers, px.(n) is o b t a i n e d from the following recurrent relation:

we construct a M a r k o v chain to m o d e l the system behavior. F o r case C the state definition is given b y n, the n u m b e r of messages in the system. The

p k ( n ) = p k ( n - k ) + p k _ l ( n -- k ) + ... p l ( n - k ) + p o ( n - k ),

is given by:

with

(n N, d~, s ~ . . . d N, sN),

Pk(n)=0, Po ( n ) = 0,

n 0

p k ( k ) = 1,

k >0

or

state of the exact M a r k o v chain for cases A a n d B

k<0

(12)

w h e r e n N is the n u m b e r of messages i n t r a n s m i s ion, d, is the d e s t i n a t i o n n o d e referenced b y n o d e (source) i, a n d s~ is the state of n o d e i. n~ can take values 0 - 2 , a n d d~ can take the

Table 1 Equations for the exact lumpable chain of a N = 4 system in case B. P0000 = Poooo"%,o + Plooo"70n + Plloo" 70.2 P~ooo= Poooo"7Lo + Plooo"7L1 + PHoo"7L2 + P2000'70,2 + Primo"70,3 + P2100"70,3 7

7

7

5

7

2

PHoo = Poooo" ~72,0 + P~ooo" 9~2,1 + P~oo" 9~2.2 + P2000" ~71.2 + P1110" ~71,3 + P2100" ~'/r1.3 + P1111 " 70.4 q- P2200" '7/'0,4 + P2110"

2

2 2 2 1 2 1 1 P2000 = PO000" ~ 2 . 0 + PlO~O" ~72,1 +/011OO" c~'r2.2 + /02000" ~71,2 + P l l l O " 971.3 + P2100" 371.3 + P300~)" 70.3 + P2110" ~¢r + P3100" 70.4 --

11

11

4

PN~o - Polo" :773,0 + Plooo")~73.1+ P2000"972,2 15e//. _~_ P2100 = P0000" ~ 3,0 Pl000" I P3000 = P0000" ~q~r3,0 + PI000"

15 5 2773,1 + P2000' ~72,2 + P3000" 71.3 1 2773,1

3

PHH = P0000"~rr4,0 4

P22oo = Poooo"2774,0 16

P2100 = Poooo"2774,0 Poooo+ P1100 + P2000 + PHlo + P2100 + P3000 + PIll0 + P2200 + P2no =1 The set of linear equations is solved by Gauss elimination.

I. Chlamtac, O. Ganz / Interfaces for Multiple Bus Networks

values 0 (no message to transmit) and k (k-th node as destination), s i can take the values 0: active (no message to transmit), - 1 : receiving a message, 1: transmitting a message, 2: transmitting and receiving a message (case B). An increase in the number of nodes, N, cornplicates the Markov chain which makes the solu: tion not easy to handle due to explosive number of states. The total number of states is: Case A:

N A = 1 + Y'. A ( i ) . i=1 C a s e B: N B = 1 + ~ B(i)

i=1 where A ( i ) and B ( i ) are the number of states at level i for cases A and B respectively as derived in Appendix 2. We therefore introduce, in the next section, approximations which reduce the size of the

181

the system has reached its equilibrium. Let Pn be the probability that there are exactly n messages in the system at the end of a service time interval and let ~r,j be as defined in (10). We define R , , the probability that among the n messages in the queue there are at least two messages whose transmission will succeed. We denote these messages by the pairs (S 1, D1), (S 2, D 2) where S i stands for the source of the message and D ~ for its destination. Using this notation we can compute for each case R,, - the probability of successful, transmission for both messages (no conflict) given there are n >/2 messages in the queue. C a s e A . The processor can transmit or receive a message simultaneously. Given the source and destination of the first message is (S 1, D 1) the second message (S 2, D 2) must be chosen D t 4:D 2 and S 1 ~ D 2 and S 2 ~ D 1 no message ] suitable for transmission ]

R,=l-p

( pA,~ f n°suitable

=1-

~

k messages

D 1 has nothing / to transmit

Markov chain and permits us to attack the problem for any N .

. ( o ~ has nothing ] x Pc ~ to transmit [

4. Approximate M o d e l s

~- pa~ messages

( no suitable

)

D1 " a ~ ( DI has a transmit nas /~ p ,"~/ ~ message " \ to transmit

t

J

n-I

The model that we introduce in this section analyzes the system behaviour by using Markov models that transition between the states containing only partial queueing information, the total number of messages in system at the end of a service timesignificantlyinterval,n. The total numbereasy to OfseeStates is thus reduced. It is that the results that we obtain in this way are therefore approximate for cases A and B, but remain exact for case C. 4.1. S y n c h r o n o u s S y s t e m

To evaluate the transition rates we make use of the independence assumption, that is we assume that at the beginning of a time interval each queued message assumes a new source/destination iden[12].tityequiprobably and independently each time To establish the set of equations we assume that

In-k-1 messagesl k messages t o / pA = ~ P ( t o d e s t i n a I destination k = 0 ~ tion 01 ] S1

J

{ k messages × P f to destination S 1 ~ 1 ( 1___~_ N 1 ),_ k- ~(\~ 1 = ~--0=

)k

=n

(~l)~-

Similarly, we obtain:

n - 1 PD=N_I ,

n- 1 Pc = 1

N-

1"

Giving finally: / 1 R , = 1 - _~~ _ )

\

• ( N n 2 - 2 n 2 - n N + N + 2 n - 1).

(13)

182

I. Chlamtac, O. G a n z / h u e f f a c e s f o r M u l t i p l e B u s N e t w o r k s

Case B. A processor can transmit and receive a message simultaneously (destination contention only), s.t. the only constraint becomes: Dr4:D2" the destination of }

( no suitable ] 1 )~ = -PA

all n messages

Rn = 1 - P ~ m e s s a g e s { = 1-

is D t

[ N*

p ( n I has a message} -0

4.2. Asynchronous

/the destination ] ] ] /of all n m e s s a g e s | ] ) + P s ( D1 has no m e s s a g e } p c ~ i s D ' / D '

hasno {

\message Ps = 1

system at the end of a unit service time interval is given by eq. (16) in Table 2. The multiple dependence on the various states prevents us from using iterative techniques for solving the set of equations (16) so the set of probabilities pi's must be solved from the set of linear equations (16). Notice that in the approximate model the size of the matrix corresponds to the number of processors.

'

}

n N

Pc = ( n/[ 1_._~/~ pA= I - N j k N _ I ] .N=(N-n).

( N I _ I ) '~

We define the system state as the number of messages in the system. Again no account is kept of the state of internal queues. The transition rates are evaluated using the averaging technique. The transition diagram in Fig. 2 is for a system with N users. For cases A and B we define a~ as the ratio between the sum of transition rates from all states with i messages to states with i - 1 messages, and the number of states with i messages in the exact chain which is derived in Appendix 2. Case A : { 1, i =

Giving finally: R, = 1 -(N-

n) ~

.

System

l

(14)

(N-2)(N-3)+

Case C: A processor can receive and transmit simultaneously or can receive 2 messages simultaneously. In this case there is no source or destination conflict:

,(N-2)(N-3)+ ~ N >~i > |

R, = 1, (15) Without loss of generality we let the service interval be equal to unity and obtain p ~ - s , the probability of the number of messages present in

,

+(N-l) ~

2

2(N-t)

,.-%-_]- + ( N - l )

Case B: We obtain 1, i = 1 N- i 1)'-' a, = i---ST + ( N - 2 ) ( N N-i , /_--Z-I- + ½( N - 2)( N - 1) ~-

N>i>l

Table 2 Equation (16) Po = P0" ~ro.0 + PJ "%.1 + P2" 7ro.2"R2 e I =

p,,[ ~l .,, + p I [

~l. ]

+ p2(~1121 R 2 + ~''.2 ( 1 1 R 2 ) ) + p ' [ R ~ l ~['.3

P,, = Po" % .o + P ~" %.1 + ... P,, + I ( % .,, + 1" R ,, + I + %. ,, + I ( 1 - R,, + 1 ) + P,, + 2" ~o.,, + 2" R ,, + 2

: PN

for n _< N l=PO''n'N

N p=l t=0

1.0+PI'~N

I.I+'"PN

l('n'2.N

I'R,v

I+%.,v

I'RN

I)+P~"n'o.,v(1-R,v)

2

1. Chlamtac, O. Ganz / Interfaces for Multiple Bus Networks

CaseC: l, i = 1 . a~= 2, 2<~i<~N.

x~~ •<

Let p, be the steady state probability for state i. We solve the system like a birth-death process: )~ = [ ) ~ ( N - i ) , 0,

183

O<~i~N,

l~

(N-l) X L _ j %

CD

(17)

"

(18)

"

~

otherwise,

/,i =/~c~i We use the formulas: Pk = P0 I-I - - ,

(19)

i = 0 /-ti+ 1

1

po =

N

k-1

(20)

)~i

1 + Y'~ ]-I k=l

i=0

1

P~ = P0 I-I X ( N - i) k = 1 ktak i N! 1--I a~ t, = P0 (P) ' ( N - i )! k = 1

0 ~
P0

Fig. 2. Transition Rate Diagram for a N users Asynchronous System.

~i+l

We substitute X, and ~+1 from formulas (17) and (18) in (19) and (20) to obtain:

(

aN~

,v N! ILI) 1 + /=x ~ ( p ) ; (N - j ) ! k=l a - 1

where p = ~/t~ is the loading factor.

5. Results Tables 3-6 compare results obtained from approximation and simulation for the synchronous and asynchronous models (cases A, and B). We see that the approximate results are so close to the simulation results that they fall within 99.9% confidence interval associated with the estimate of the average queueing delay. Figures 3-6 plot the average queueing delay versus the load factor p = 2~/# for the two systems for the three cases. For both systems and for all system sizes (N = 4,12) performance inproves as we move from case A to case C. This is, of course, expected since the amount of duplication in the reception/transmis-

Table 3 Average queueing delay comparison (case A, N = 4) approximation and simulation for synchronous and asynchronous models. X p= #

Average queueing delay synchronous

0.01 0.05 0.1 0.5 1 3

asynchronous

Simulation 99% confidence interval

Approximation

Simulation 99% confidence interval

Approximation

0.5125 + 0.0162 0.5507 + 0.0178 0.6270 ___0.0195 1.0000 ± 0.0251 1.2530 + 0.0291 1.6180 + 0.0305

0.5125 0.5501 0.6214 1.0000 1.2405 0.6028

0.0234 +_0.0021 0.1099 ± 0.014 0.2312 + 0.0157 0.8951 + 0.0238 1.2698 + 0.0251 1.6101 + 0.0301

0.0233 0.1165 0.2307 0.8994 1.2654 1.6115

I. Chlamtac, O. Ganz /Interfaces for Multiple Bus Networks

184

w Case A 1.5

1

Case

B

Case

C

0.5

I

0

0.5

i

1

t

L

1.5

2

i

2.5

i~

3

P

Fig. 3. Average Queueing Delay vs. The Load Factor p for N = 4 system. Cases A,B and C. Synchronous System. Table 4 Average queueing delay comparison (case B, N = 4) approximation and simulation for synchronous and asynchronous models. X p= /t

Average queueing delay synchronous

0.01 0.05 0.1 0.5 1 3

asynchronous

Simulation 99% confidence interval

Approximation

Simulation 99% confidence interval

Approximation

0.5042 + 0.0104 0.5209 + 0.0171 0.5415 + 0.0192 0.6749 + 0.0213 0.7772 + 0.0241 0.9401 + 0.0258

0.5042 0.5208 0.5413 0.6743 0.7735 0.9340

0.0071 + 0.0012 0.0361 + 0.0081 0.0708 + 0.0124 0.2942 + 0.0195 0.4669 + 0.0219 0.7392 + 0.0283

0.0074 0.0365 0.0713 0.2936 0.4675 0.7390

I. Chlamtac, O. Ganz / Interfaces for Multiple Bus Networks

4

/'/d"

I

0

185

i

0.5

1

I

L

i

1.5

2

2.5

p

Fig. 4. Average Queueing Delay vs the Load Factor p for N = 12 system. Cases A,B and C. Synchronous System.

Table 5 Average queueing delay comparison (Case A, N = 12) approximation and simulation for synchronous and asynchronous models. p= #

Average queueing delay synchronous

0.01 0.05 0.1 0.5 1

asynchronous

Simulation 99% confidence interval

Approximation

Simulation 99.9% confidence interval

Approximation

0.5078 + 0.0161 0.5980 ± 0.0187 0.7369 + 0.0251 3.0019 + 0.0428 3.9902 + 0.0532

0.5156 0.5925 0.7269 3.0014 3.9902

0.0301 + 0.1721 + 0.4092 + 3.0010 + 3.9975 +

0.0291 0.1698 0.4133 3.0200 4.0000

0.0069 0.017 0.0245 0.0512 0.0521

I. Chlamtac, O. Ganz / Interfaces for Multiple Bus Networks

186 W

1.5

1

Case

B

0.5

i 0

0.5

i 1

I 1.5

i 2

i~ 2.5

p

Fig. 5. Average Queueing Delay vs the Load Factor 0 for N = 4 system. Cases A,B and C. Asynchronous System.

Table 6 Average queueing delay comparison (Case B, N = 12) approximation and simulation for synchronous and asynchronous models. ;k p= /t

Average queueing delay synchronous

0.01 0.05 0.1 0.5 1

Asynchronous

Simulation 99% confidence interval

Approximation

Simulation 99.9% confidence interval

Approximation

0.5007 + 0.0164 0.5596 + 0.0180 0.6796 5:0.0205 3.0020 + 0.0446 3.9990 + 0.0541

0.5067 0.5555 0.6637 3.0010 3.9901

0.0113 + 0.1014 + 0.3066 + 2.9912 + 4.0000 +

0.0116 0.0963 0.2999 3.0090 4.0000

0.0036 0.0116 0.0198 0.0448 0.0482

I. Chlarntac, O. Ganz / Interfaces for Multiple Bus Networks

187

w Case

0

0.5

1

B

1.5

2

2.5

p

Fig. 6. Average QueueingDelay vs. The Load Factor p for N = 12 system. Cases A,B and C. AsynchronousSystem.

sion mechanisms increases from A to C. For both systems we find that the larger the system size the smaller the additional advantage of BIU duplication in terms of average queueing delay. This trend is well supported by intuition since with the given uniform reference model, the larger the system size the less noticeable become the source conflicts (case A) and the destination conflicts (cases A,B). In other words, for large N the bus contention becomes the dominant factor. It is also interesting to note that for small systems the duplication involved in going from case A to case B (i.e., eliminating the source conflict) has significantly more noticeable effect on improving performance than has the duplication involved in going from B

to C (i.e., eliminating also the destination conflict). Thus, in terms of delay performance alone, the additional cost of BIU logic duplication involved in going from B to C may not always be justified.

6. Summary We have described a distributed communication system with a two bus communication system. We have further proposed several alternatives for the processor bus communications interface (BIU) design. An analysis of the various models based on Markovian models has been provided. It was shown how exact models can be con-

188

1. Chlamtae, O. Ganz /Interfaces for Multiple Bus Networks

structed for synchronous and asynchronous systems. We have also provided a computationally more efficient Markov model which serves as excellent approximation, The study has demonstrated the effect of BIU design decisions on the performance of the multiple bus communication system. It was shown that with the small to medium size systems which are most common today the incorrect design of the communication interface can be the major source of performance degradation, The presented results should thus be relevant in studying the behavior of distributed systems and can aid in making application oriented perform a n c e / c o s t decisions.

tion rates for the approximate asynchronous systern. The level of a state is defined as the number of messages in the system. We assume N >/3. At level 0 there is only one state. Case A . There are two cases for nu the number of messages in transmission: a) n N = 1. If the message transmitted is (S 1, D 1) then there are two subcases: (i) If D 1 is a source then the destination of i - 2 messages is S 1 or D a. The number of states in this case is

(N - 1)(N--22)2i-2 where ( N - 1 )

is the number of ways D 1 can

choose its destination, ( y 2 2 ) is the number of

Appendix 1 We calculate the average number of messages in system L. L is the expected number of messages in the system at the beginning of a service interval and since packets cannot leave the system during the service interval we add the integral of the

ways we can choose i - 2 sources for the i - 2 messages and 2 ~ 2 is all the possibilities of the i - 2 messages choose their destination (S ~ or D1). (ii) D ~ is not a source. Then the destination of i - 1 messages is S 1 or D 1. Similar to i) the number of states in this case is

expectedinterval. service number of message arrivals during the

(/N_-12 ) 2i- 1

N

L = j g= l jpj +

The message transmitted can be chosen among N ( N - 1 ) possible messages, so the number of states for n N I is:

f," e (x,)d, t=0

N

=

= ~

£i

j=0 N

N~.j

(l_e_X,

+(N-l)

2 ~-2 .

i=0

= Y'..pj(N

- j ) ( 1 - e -xt)

b) n N = 2. There are at least two suitable messages for transmission. The number of states is:

j=0

N

N

j=l

t=0j=0

\t---Z

L= Ejpj+f T Epj(N-j)(1- e_At ) dt Assuming the service interval equals unity ( T = 1) we obtain: N - ~_, jp/

.

N(N-

1)=

the number of possibilities of choosing the first transmitted

( N - 2)(N - 3) = the number of possibilities of

j=0

choosing the second messages; the order ,between the first and the second message is not important;

Appendix 2 In this appendix we give the expression for the number of states at level i and derive the transi-

l

X ( N - - l) i-2"

=

the number of ways to choose the source of i - 2 messages;

L Chlamtac, O. Ganz / Interfaces for Multiple Bus Networks

(N-1)

the n u m b e r of ways to choose the destination of i - 2 messages. Therefore the n u m b e r of states at level i is: i-2=

A ( i ) = A 1( i ) + A 2 ( i ) .

Case B.

a) n N = 1 all the messages have the same destination. The n u m b e r of states is ( N- 2 ] B 1( i ) = N( N - 1) ~ i - 1 ! " b) n N = 2 there are at least two suitable messages for transmission• Similarly to case A we obtain /

B 2 ( i ) = ½ N ( N - 1 ) [ ( N - 1 ) ( N - 2) ( •

N i - - 22

)

(N-l)

`-2

)

and therefore B ( i ) = B 1( i ) + B 2 ( i ) . Using the averaging technique the transition rates a i are: for case A: a, =

A 1(i) + 2A 2 (i) Al(i ) +A2(i)

[ 2 ,,-2[ N-i ) (N-Z)(N-3)+~Z--~) ~2. _--Z i i- + ( N - l ) / ( 2 )i-E( )' = ] 2t (N - 2 ) ( N - 3)+ ~ 2./_~N-i + ( ' N - 1) 1,

N >1i > 1, i =1;

for case B: B a ( i ) + 2 B 2 (i) a, =

B l ( i ) + B2(i )

--~ . . . . . +7(N-2)(N~1,

-1"~ )

,

N>~i>l, i=1.

189

References [1] H.A. Freeman and K.J. Thurber, "Microcomputer networks", IEEE Computer Society, 1981. [2] K.J. Thruber, H.A. Freeman, "Local Computer Networks", IEEE Computer Society, 1981. [3] W.R. Franta, J.R. Heath, "Performance of Hyperchannel Networks: Parameters, Measurements, Models and Analysis", T R # 82-3, January 1982, University of Minnesota. [4] W.R. Franta, J.R. Heath, "A Simple Model of Hyperchannel local Network Channel Selection Procedure in MultiChannel Networks", TR, 1982, University of Minnesota. [5] Vax Cluster Systems: Overview, Digital Equipment Corporation, April 1983. [6] M.A. Marsan and M. Gerla., "Markov Models for Multiple Bus Multiprocessor Systems", IEEE Trans. on Com-

puters, Vol. C-31, No.3, March 1982. [71 M.A. Marsan, D. Raffinella., "Nonpersistent M-CSMA Protocols for Multichannel Local Area Networks", 7th Conf. on LCN MPLS, October 1982. [8] M.A. Marsan., "Multichannel Local Area Networks, Compcon, Fall 1982. [9] G.D. Henley and T.F. Fiorino., "Avionics/Navigation Architectural Design Considerations", NTC, Galveston, Texas, November 1982. [101 R. Flower, "Q Buss Contention in a Multiprocessor System", Internal DEC Memorandum, August 1982. [11] w.c. Hohn, "The Control Data Loosely Coupled Network Lower Level Protocols", Proc. of NCC, May 1980, pp. 129-134. [12] J. Gadre, T.E. Stern, "A Comparison of Multiple Access Protocols for Packet Switching in Satellite Switched Multibeam Systems", in Conf. Rec. Int. Conf. Commun., Seattie, WA, June 1980, pp. 58.4.1-58.4.5. [13] A. Tanenbaum, Computer Networks, Prentice Hall, 1981. [14] J.G. Kemeni and J.L. Snell, Finite Markov Chains, New York: Van Nostrand, 1960. [15]W.R. Franta, andM. Bilodeau,"AnalysisofaPrioritized CSMA Protocol Based on Staggered Delays", Acta Informatica, 13 (1980) pp. 299-324. [16] O. Spaniol, "Analysis and Performance Evaluation of Hyperchannel Access Protocols", Performance Evaluation, 1981, pp. 170-179. [17] K.J. Thurber, H.A. Freeman, Architecture Considerations for Local Computer Networks, 1-st Intermational Conf. on Dist. Computing Systems, September 1979; also Tutorial LCN, second edition Computer Soc. Press, 1981.