A Queueing Model of a BufferlessSynchronous Clos ATM Switch withHead-of-Line Priority and Push-Out

Arne A. NilssonFuyung Lai

Center for Communications and Signal ProcessingDepartment Electrical and Computer Engineering

North Carolina State University

TR-90j21December 1990

A Queueing Model of a Bufferless Synchronous ClosATM Switch with Head-Of-Line Priority and

Push-out *

Arne A. Nilsson and Fuyung Lai

Department of Electrical and Computer Engineerillg, and

.Center for Communications and Signal Processing

North Carolina State University

Raleigh, N.C. 27695-7914

Harry G. Perros

Department of Computer Science, and

Center for Communications and Signal Processing

North Carolina State University

Raleigh, N.C. 27695-8206

1990

Abstract

We consider a synchronized bufferless Clos ATM switch with input cell processorqueues. The arrival process to each input port of the switch is assumed to be bursty andit is modeled by an Interrupted Bernoulli Process. Two classes of cells are considered.Service in an input cell processor queue is head-of-line without preemption. In addition,push-out is used. That is, a high priority cell arriving to a full queue takes the spaceoccupied by the low priority cell that arrived last, as long as the cell is not in service.Each cell processor queue is modeled as a priority IBP /Geo/l/K+l queue with push­out. We first present an exact analysis of this priority queue.The results obtainedare then used in an approximation algorithm for the analysis of the ATM switch.Validation tests using simulation data shows that the approximation algorithm has agood accuracy.

•Supported in part by DARPA under grant no. DAEA18-90-c-0039.

1 Introduction

Broadband ISDN is a promising communications infrastructure for the future. It is conceivedas an all-purpose digital network, and it will provide an integrated access that will support awide variety of applications for the customers. The Asynchronous Transfer Mode (ATM) hasbeen strongly promoted as the target transfer mode solution for broadband ISDN. ATM isa high-bandwidth, low-delay, packet switching technique, capable of efficiently multiplexinga number of highly bursty sources, such as voice, file transfer, and video.

In recent years, several types of ATM switch architectures have been proposed. Oneimportant class of architectures is based on multi-stage interconnection networks. There arebuffered or unbuffered switching elements in a multi-stage interconnection network. In theunbuffered case, there may be buffers at the input ports or at the output ports of theswitch. This type of structures have been reported in Turner [1], Narasimha [2], Huangand Knauer [3], Giacopelli, Littlewood, Sincoskie [4], and Tobagi and Kwok [5]. Someswitch architectures provide full connectivity between the input and output ports, such asthe bus-matrix switching architecture [Nojima [6]), the knockout switch (Yeh, Hluchyj, andAcampora [7]), and the integrated switch fabric (Ahmadi [8]).

Other architectures that have been proposed are the shared memory and the sharedmedium architectures. In the shared memory architecture, a single memory is shared by allinput and output ports. There is a single controller to process the incoming and outgoing cellswhich are stored in the same memory. Examples of this type of architecture are discussedin Devault, Cochennec, and Servel [9], and Kuwahara, Endo, Ogino and Kozuki [10]. In theshared medium architecture, arriving cells are multiplexed onto a parallel bus. An exa.mpleof this architecture is discussed in Suzuki [11]. A good review of ATM switch architecturesis given in Tobagi [12]. Approximate analytic studies of ATM switches can be found inKarol, Hluchyj and Morgan [13], Hluchyj and Karol [14], Patel [15], Yoon, Lee, and Liu[16], Shaikh, Schwartz, and Szymanski [17], Oie, Suda, Masayuki, and Miyahara [18], andYamashita, Perros and Hong [19].

Different classes of service provided by an ATM network will require different qualityof service. In particular, voice and video are tolerant to cell loss but not to time delays. Onthe other hand, the transfer of bulk files is tolerant to time delays but not to cell loss. Inview of this, it has been proposed to introduce priorities among cells. These priorities areknown as space priorities, as they deal with priorities regarding the utilization of the spacein a buffer. Space priorities can be also used in conjunction with congestion control at thecall level.

Two different space priority schemes have been proposed, namely, push-out andpartial buffer sharing. In these two schemes, there are high or low priority cells. In thepush-out policy, high and low priority cells are admitted to the buffer as long as there isspace available. A low priority cell is lost if it arrives to a full buffer. A high priority cell that

1

arrives to a full buffer, takes the place occupied by the low priority cell that arrived last , aslong as this cell is not in service. A high priority cell is lost only if there are no low prioritycells in the buffer waiting for service. In the partial buffer sharing scheme, both high andlow priority cells are admitted to the buffer as long as the total number of cells in the bufferis below a pre-defined threshold. Once the threshold is exceeded, only high priority cellsare admitted to the buffer. Garcia and Casals [20] analyzed a finite capacity single serverqueue with two classes assuming partial buffer sharing. The arrival process was modelledby a two-state Markov Modulated Poisson Process, and the service time was assumed to beconstant. A similar model was considered by Le Boudec [21] assuming a number of identicalarrival streams, each being modelled by a two-state Markov Modulated Bernoulli Process.Kroner, Hebuterne, Boyer, and Gravey [22] compared the two space priority schemes, Theyconcluded that both schemes have a comparable performance, though the partial buffersharing scheme is easier to implement. Their results, however, have been obta.ined underthe assumption of Poisson arrivals. Gravey and Hebuterne [23] analyzed an M/G/1/n queuewith head-of-line non-preemptive priority and push-out. The partial buffer sharing schemewas also considered by Yin, Li, and Stern [25]. Conservation laws for the push-out schemewere given in Sumita and Ozawa [24]. Finally, for a literature review of ATM related papers

see Perros [26].

In this paper, we study a blocking ATM switch with finite buffer at the input ports.The switch fabric is a bufferless Clos three-stage interconnection network. The arrival processto each input port of the switch is assumed to be bursty, and it is modelled by an InterruptedBernoulli Process (IBP). Each input queue is assumed to have space and service priority. Inparticular, we consider push-out for space priority and head-of-line with no preemption forservice priority. We now proceed to describe the switch under study in detail.

2 Model description

Let us consider an N x N Clos cell switch, as shown ill Figure 1. There are three stages inthe switch. At each stage, there are n or VN number of bufferless switching elements. Eachswitching element is an n x n crossbar switch. For each input port, there is a finite queueserved by a cell processor. There are two classes of cells, high and low priority. If an arrivinghigh priority cell finds the buffer of the cell processor full, it will push out a low prioritycell with the least waiting time in the system, but not the one in service. An arriving lowpriority cell is lost if it finds the buffer full. An arriving high priority cell is lost only if thebuffer is full with high priority cells. High priority cells have also priority for service over

low priority cells. The service priority is head-of-line with no preemption.

The cell processor is responsible for determining the output port of a cell andthen transmitting the cell through the switch fabric. The operation of the C10s switch issynchronous. That means, at the beginning of a time slot all the busy cell processors will

2

1 -..[l]IJ- 12 .

1 2.0

n -..0IIl- n

n+l Sf n+l

2n 2n

00

0 0

0 0

N-n+l EflN-n+l. . . 0

0 n • n 0 00

-+-OID- 0 0 0

N N

Figure 1: Synchronous N x N Clos ATM Switch.

3

launch a cell through the switch. A cell will successfully get through the switch if it finds afree path to the sought output port. No contention takes place at the 1st stage of the ClosATM switch. The cells in the switch will compete for the output links at the 2nd and the3rd stages. In a switching element, if more than one input link compete for the same outputIink, one of the input links will get through. This input link is randomly selected. If a cell isblocked in the switch, the switching element notifies the cell processor, and transmission ofthe cell is aborted. The cell processor will retransmit the cell at the beginning of the nexttime slot. The retransmissions will continue until the cell is successfully transmitted throughthe switch.

The time required for a cell to pass through the switch fabric is equal to one slot.It is assumed that each incoming link is slotted. The length of a slot is equal to a slot ofthe ATM switch. Each incoming link is not synchronized with the other incoming links andwith the switch. A cell that arrives at an idle cell processor in the middle of a slot of theATM switch is not transmitted until the beginning of the next time slot of the ATM switch.

The arrival process to each input port of the ATM switch is modeled by an Inter­rupted Bernoulli Process(IBP), as shown in Figure 2. No arrivals occur if the process is inthe idle state. If the process is in the active state, arrivals occur in a Bernoulli fashion. Thatis, a slot will contain a cell with probability cx. If the process is in the idle (active) state,then in the next time slot it remains in the idle (active) state with probability q (p) or it willchange to the active (idle) state with probability 1- q (1- p). It can be shown (see Nilsson,

1-p

p

1-q

q

Figure 2: TIle Markov chain of the IBP process.

Lai, and Perros [27]) that the z-transforrn of the probability distribution of the interarrival

time is

A(z)Z Q [p + Z (1 - p - q)]

(1 - a)(p +q - l)z2 - [q +p(1 - a)]z + 1·

4

From this we have that the mean interarrival time E(i) is equal to

E{l} =(2 - p - q)

0:(1 - q) ,(1)

and the squared coefficient of variation of the interarrival time, C2, is

02 = 1+ a ((1 - p)(3 - q) _ 2) +a? (1 - q)2 .(2-p_q)2 (2-p_q)2

(2)

(3)

1 - 0- =

Assuming that the destination of a cell is uniformly selected, it can also be shown that theprobability for a cell successfully passing through the Clos ATM switch ,1 - (T, is

Average number of busy output links at the 3rd stage

Average number of busy input links at the 1st stageN

" [3]LJPii=l

N

LPii=l

where p~3] is the ith output link utilizations of the switching elements in the 3rd stage of theClos ATM switch.

In this paper, we analyze the buffer in front of a cell processor as a two-classIBP /Geo/l/K+l queue with head-of-line non-preemptive priority and push-out. In section3, we obtain the queue-length distribution of the number of high and low priority cells thatare waiting to be served, as seen by an arriving cell. We also obtain the distribution of thewaiting time of a high priority cell. In section 4, we obtain the probability that a low prioritycell will get served given that it entered the buffer. We also compute the mean waiting timeof a low priority cell given that it entered the buffer and it got served. In section 5, weanalyze the ATM switch, following the simple approximation algorithm described in Nilsson,Lai, and Perros [27]. Finally, numerical results are given in section 6 and conclusions aregiven in section 7.

5

3 The IBP/Geo/l/K+l queue with head-of-line pri­ority and push-out

Let us consider a cell processor queue. It is obvious from the previous section that the totaltime it takes for the cell processor to transmit successfully a high or low priority cell throughthe switch fabric is geometrically distributed with probability a . Therefore, a cell processorqueue can be modeled as all IBP /Geo/ 1 queue with finite buffer, head-of-line priority, andpush-out , as shown ill Figure 3. In the analysis below, we will consider this queueing system

IBP

Figure 3: The IBP /Geo/l queueing system with finite buffer.

at successive arrival points as shown in Figure 4. These arrival points are the imbeddedMarkov points of the queueing system. Our approach is to focus attention on arrival instantsand the number of high and low priority customers seen by an arriving customer.

(m,n) So~ .0 _ 0 • • 0" 0 __.J

(i,j)

I slots

Figure 4: The evolution of the number of customers in the system.

Below, we distinguish between the queue that is allowed to be built in front of theserver and the system that includes the queue and the server. Let K be the queue size, andK+1 be the system size. Let (i,j) be the state of the queue as seen by an arriving cell, wherei is the number of high priority cells and j is the number of low priority cells in the queue.The state (0,0) means that there are no cells in the queue. Finally, state °means that the

entire system is empty.

In order to analyze the queueing system we introduce the following notation

6

{3 : the probability that an arriving cell is a high priority cell.

Pi,; : the probability that an arriving cell finds the queue in state (i,j).

Po : the probability that an arriving cell finds an empty system.

i : the interarrival time of the cell.

X : the service time of a cell.

We define

and

p[i == l] 1> 1

P[x == i] == (1 - U)Ui-

1, i ~ 1,

where al is the probability that an interarrival time is 1 slots, and a is the probability thata cell is blocked in the C105 ATM switch.

Below, we obtain the probability Pi,; and we also obtain the waiting time distribu­tion of a high priority cell.

3.1 TIle queue length distribution Pi,i

We first note that in Nilsson, Lai, and Perros [27], the steady state probability 'i that anarriving cell sees j cells in the system was obtained for an IBP /Geo/l queue with finitebuffer. We observe that

Po ,0,Po,O /1·

Let us define

bo == A(u),

where A(z) is the z-transform of the probability distribution of the interarrival time. III

order to obtain Pi,i we need to study the system at instants of arrivals. In steady state, wehave

p..1.,3

a K~-j P, .~ ( 1 ) ul-(n-i+1)(1 _ )(n-i+l)fJ ~ n,3 Z:: . + 1 a aln - ~

n=i-1 [=1

7

and

K -1 00 ( 1 )f3 :E Pn,o:E . U'-(n-i+l)(l - U)(n-i+l)al

n=i-l l=1 n - ~ + 1

+f3(PK,o + PK-1,d f ( K ~ i ) ul-(K-i)(l - u)(K-i)acl=1

/3(Pi-1,obo + Pi,ob1 +... + PK-1,obK- i)+(3PK,obK- i + fjPK-1,lbK - i

for K > i > 0, and j = 0,

and

K-l-(j-l) 00 ( 1 )Po,i (1 - (3) E t; Pn,i-l n u/-

n(1- u)n a/

K-l K-l-n 00 ( 1 )+ L. L L Pm,n m + 1 +n _. u/-(m+l+n-i)(1 - u)(m+l+n-i)a/

n=1 m=O l=1 . J

+f3 t PK-n,n f ( K ': · ) ul-(K-i)(l - u)(K-i)a,

n=;+l 1=1 J

+(1 - (3) t PK-n,n f ( K ': . ) u'-(K-i)(l - u)(K-i)a,n=j [=1 J

(1 - (3)(PO,i-l bO + P1,i-lbl + ... + PK-i,i-lbK-i)

+Po,ibl + P1,jb2+.,. + PK-l-i,ibK-i) +(1 - (3)PK-i,i bK-i

+PO,i+lb2 + P1,i+lb3 +..,+ PK-l-(i+l),i+lbK-i) + PK-(i+l),i+lbK-i

+...+PO,K-1bK-i + P1,K-1bK - i

+Po,KbK - j

for i == 0, and K > j > 0,

For the case i +j == K, we have00 00 00

PK,O = PK,O L u/a/ + f3PK- 1,o L u/a/ + f3PK-1,lL u/a/l=1 l=1 l=l

8

and

00 00

PO,K (1 - /3)PO,K :L eT'a, + (1 - /3)PO,K-l L: eT'a,1=1 1=1

and

00 00

f3Pi-1,K-i L: utal + j3 P i - 1,K - i+1 L:Utal +1=1 [=1

00 00

(1 - (3)Pi,K-i L ulal + (1 - (3)Pi,K-i-1 L ulal

1=1 l=1

for K > i > o.

From the above equations, we obtain the transition probability matrix of the system shownin Figure 5. Thus, the steady state probability, Pi,j, can be computed using a numericalmethod.

9

a

a

....... J3bo......... PhoJ3b

oPbo

o

o

o

o

o

o

o

o

,,,,

,

o

o

o

o

o

o

o

o

o

o

o

o

~ ~ :;J ~ :J' ::P J1 ?! r ;sJ.~ 'f f! " :;0 ~:;'~ :F .:0o : 0 : 0 ~: 0 ~ '" : • t: ~ ~ ~ ~ • t"' i: ~ ~ • 0 t"' ~ ~". • • 0...., W '" • 0 ...., "'....,....., ....

- . . -:.- .. i - . - -- .. ~. - . - . - . - - -- . - ~ - . - - - -. --.. -..~ - -. -- - -- --- . - _. __ . :._ _. _ __ .. : . _.. . _. _._.. _1:0: 0: 0 : 0 : 0 : 0 : a

....: i-······~············· ~ _..~ - ~ _ _ : _ _ _0: 1: 0: 0 : 0 : 0 : a : a

•••• ~Ah-:-ti,;···· .:-l'.a~•••••.... .:.••.•..•....•...• ~ ••...•.....•••••.•••••.•• ':'a~······················ .It_·· •• l:fl=\••••••••••••••••••••••:..,-, :"'-1 a •~ 0 a : :~ a 0 :1-""': 0 0 :0 I-""': 0 0o :_ 0:_ : _ 2 : • .... :_ K-2 :_ K-I : K-I_ K-I:Ah :Ah b • Ah b b : .Ah b u __ b .Ah b· b : o. Ah b b• ,.,....0.,.,....1 1:,.,....~ 2 ? ''''-K'-? K'_? K-2. """'K-I K-l x-.i: ..,....F~-I K-l K-l...T· "1i3b

0"O·:~I···O···~"·r······· ..··.. ···r~~~3·~· ;.~.::::::: ... ~. ··1~K··:··O··· ..:.:.:..0·" !bK·~ ·2·~~·"'···0···::::::::· 0···

..: • , K- • -.... • ...• 0 'Ah'- • '- .- , Ah

0: pJ....o(3bo:J3b1 J3b1 0 ~ ~~-3 ~-3 0 ... 0 ~J3bK-2 ~-2 0 ... 0 ~ 0 V'""K-2 J3bK- 2 0... a: : 0 Ah: 0 ~ b : : Ah b··· b : 0 Ah b··· b : 0 0 'Ph b··· b: : """'0: 1 1: : 0 tJ"""K-3 K-3 K-3: fJ"""K-2 K-2 K-2: K-2 K-2 f~-~

····I····r····.··~..·· · " _ _::: :: : :::: :: : :: : 0 : ',: : : :

:: : ':: : :::: :: : :'" '. . .~ :..---.. .:. ----- u: -- - !pli'.i;_u.--.. --.-----'.t~··0"·-- -- ---- ~.b--j3i:i' --0- -- ---.,.,

• 1 • 2 2 2

:i3b1 I3bI 0 1;3b2 ~ 0 0 ;3b2 ~

PhI ' 0 ;3b., \ ~2••. '" ', ......... .Aha a \~OO 0 \~OO"""'1 " 2 \ 2

PhI bl , ';3b2 ~ ~. i3b2~ ~.... ····r·······~············· -:- .

~ J3b0 ~ f3b1 0 • b1 13b1 0

';3b0131'0 0 ~Phl J3b1 0 0 ~l J3b1, .i3bo """ ~ 0 PhI ' Ph1-. : ,"

........ ' ''''oAh: .......... """"0 : \

...... PhoJ3b0

~ 0 J3b1 0

Pho~ phI b1 :. .... ~ ··························T~··o··················· ·b~···I3b~················_····

,Pho

l3bo

0 0 i3bol3bo

0

o i3bo .... ;3bo..............

................,

Poa~·"

PI;·"

POI

P20

PI I

P02

P1 , K- 3

PO, K- 2

PK-I,O

*P=l-PFigure 5: The probability transition matrix of the system.

10

3.2 The Waiting Time Distribution of the High Priority Cell

Let Xo be the time interval between the arrival instant and the beginning of the next slot,and let w be the waiting time in the system. We have

w=

Xo

1 + Xo

2 + ~o

3 + Xo

K - 1 + Xo

, l-~K.O [Po +Po,o(l - u)]

[K( l ) K-l(l) . ]

'l-~K.O .r; 0 PO,ju(1 - u) +.r; 1 PIji - u)(l - u)

[

K -IK-i ( K _ 1 ) . ]'l-~K.O ?=?= i pi,jUK-1-'(1 - u)i(l - u)

t=O 1=0

The distribution of the waiting time is as follows:

l-~K.O [Po + Po,o(l - u)] , n = O.

P[w = n + Xo] =

n>I(-l

11

The mean waiting time of a high priority cell can be obtained by the followingargument. Let Pi~ be the probability that an arriving high priority cell finds the system instate (i,j) given that the high priority cell will be accepted into the system and let NQh bethe average number of high priority cells in front of the arriving high priority cell. We get

NQ h ==K-l K-i

~ '" ipfl.L..J L..J "',3i=O j=O

1 K-l K-i

1- P L L iPi,jK,O i=O j=O

The residual service time ltVo seen by the arriving high priority cell is found as follows:

Wo == 1 [xoPo + (1 - Po - PK,O) f(xo + i)(l _ (j)(ji]1 - PK,O i=O

1 [xo(l _ PK,O) + (_0'_) (1 - Po - PK,o)]1 - PK,o 1 - 0'

Thus, WH, the mean waiting time of tile high priority cell is :

1ltVH == Wo + NQh - - ·1-0'

12

4 Computation of the probability that a low prioritycell will get served and mean waiting time.

In this section, we obtain exact results for the low priority cells. In particular, we calculatethe probability that a low priority cell will get served given that it entered the queue. Wealso obtain the mean waiting time of a low priority cell given that it entered the queue andit got served. The main vehicle for obtaining these results is the one dimensional randomwalk consisting of the absorbing states 0 and K+l, as shown in Figure 6.

o 1 2 3 K-1 K K+1

mJ :Good absorbing state ( cell gets service).

ttl : Bad absorbing state (cell gets pushed out of queue).

Figure 6: The states of the random walk.

TIle states of the ra.ndom walk are defined as follow. We tag a low priority cellwhen it enters the queue. The state variable of the random walk indicates the position ofthis tagged cell throughout its residency in the queue immediately after a high priority cellarrives. We note that the positions in the queue are numbered from 1 to K. The positionin front of the server is numbered o. The tagged unit receives service if the random walkreaches the absorbing state 0, and it is pushed-out if the random walk reaches the absorbingstate K+l. If upon arrival the tagged unit sees i high priority cells and j low priority cellsin the queue, the random walk will start at state i+j+l. Transitions will then occur onlywhen a high priority cell arrives. We define our transitions in this way, seeing that only thearrival of a high priority cell may affect the position in the queue of the tagged unit. Thesetransition probabilities are calculated below.

13

4.1 Computation of the transit ion probabilities

Let Pi --+ j be the transition probability from state i to state j. Let S, be the probability ofhaving i cells served during the interarrival time of the high priority cell. Let P[t a == n] bethe probability that the interarrival time of the high priority cell is n slots and 1 - a be theprobability of having a service completion in one slot. Then, for j > 0, we have

Pi --+i P[(i - j + 1)service completions]

Si-i+1

f ( i _ ~+ 1 ) (1 - u)i-i+1u n-(i-i+1)P[t a == n]n=O J

(1 )i-i+l 00 ,- (j L n. (jn-(i-i+l) P[t = n]

(i-j+l)! n=i-i+l (n-(i-i+l))! a

(1 - (j )i-i+l T(i-i+ 1 ) ( (j)

(i-j+l)! a

and, for j == 0, we have

i-I

Pi -+O == 1 - L Sii=O

where TJi-i+1 ) is the (i + j - 1)th derivative of Ta(z) at z == a , and

00

Ta(z) == L znp[ta == n].n=l

The above generating function is computed as follows. Let t a be ~he t.ime interval from. anactive state to an arrival of the high priority cell and ti be the time interval from an Idle

state to an arrival of the high priority cell. We have

{1 ,po.(3

1 + ta ,p(l - 0.(3)

1 + ti ,1- P

{1 + ti ,q

1 , (1 - q)a(3

1 + t a ,(1 - q)(l - 0.(3)

Hence,

Ta(z) zpaj3 + zp(l - aj3)Ta(z) + z(l - p)Ti(z)

Ti(z) zqTi(z) + zaj3(l - q) +z(l - q)(l - aj3)Ta(z)

14

We get

Ta(z)za,B[p + z(l - p - q)]

(1 - a,B)(p + q - 1)z2 - [q +p(l - a,B)]z + 1

4.2 TIle probability that a low priority cell will get served

Let U, be the probability that the random walk system reaches the absorbing state 0 giventhat the system starts in state i. Let Sj be the probability of having j cells served ill aninterarrival time of the high priority cell. We have

Uo=landUK +1=O.

To find Ui, i = 1,2,· . · ,K, we have to solve the following linear equations :

U1 SoU2 +(1 - So)Uo1

U2 SoU3 + SlU2 + (1 - L SI)UO

1=0

K-2

UK - 1 SoUK + SlUK - 1 + ... + SK- 2U2 + (1 - L Sl)UO

l=O

K-l

UK SOUK+ 1 + SlUK +... + SK- 1U2 + (1 - L SI)UO. l=O

Thus, we get:

o

+K-l

(1- L SI)UO

[=0

The above system of linear equation is solved numerically ill order to obtain U2 , U3 , · · · , UK.Using U2 and Uo we can also calculate U1 • Thus, P6~r1Jice' the probability that a low prioritycell will receive service given that it entered the queue is:

K-IK-l-·i

Po + L L Pi,jUi+j +1i=O j=O

K-IK-l-i

Po+ ~ ~ p ..c: c: '1.,3i=O j=O

P6~rvice == ----------

15

4.3 Mean waiting time of a low priority cell

Let Po(n, i) be the probability that the random walk system reaches absorbing state 0 afterexactly n high priority arrivals given that it sta~ts in state i. We have

Po(O, i)i-I

1- LSi;=0

i+1

Po(n, i) L Po(n - 1,j)Si+l-i;=2K

Po(n, K) = L Po(n - 1,j)SK+l-jj=2

for n 2: 1 and 1 S i < [{

Let No( i) be the expected number of high priority cells that arrived before the system reachedabsorbing state 0 given that the system starts in state i. We get

for i == 1,2,···,K -1,

00

No{i) = L nPo(n, i)n=Oi+1 00

L L nPo(n - 1,j)Si+I-;j=2 n=1

f. Si+l-i {~(n - l)Po{n - 1,j) +~ Po{n - 1,j)}

i+l 00 i+l 00

L Si+l-j L nPo(n,j) +L Si+l-j L Po(n,j)j=2 n=O j=2 n=O

i+l i+lL Si+l- j No(j ) + L Si+l- j Uj

;=2 ;=2

and

00

No(K) = L nPO(n, K)n=O

K 00

L L nPO(n - 1,j)SK+I-;j=2 n=l

t. SK+l-i {~(n - l)Po{n - 1,j) +~ PO{n - 1,j)}

KooK 00

:E SK+l-i :E nPO(n,j) +~ SK+l-i :E PO{n,j)j=2 n=O J=2 n=O

16

K K

·L SK+l-jNo(j) + L SK+l-jUj;=2 ;=2

To find No(i), i == 1,2,· · · , K, we solve the following linear equations numerically:

No(l)

No(2)

3

SlNo(2) + SoNo(3) +L S3- j Uj

j=2

K

No(K - 1) SK-2 NO(2) + SK-3 NO(3) + ... + SoNo(K) + L SK-jUj

j=2

K

No(K) = SK-1No(2) + SK-2 NO(3) +... + SlNo(K) +L SK+l- j Ujj=2

Thus, No( i), the expected number of high priority cells that arrive before the tagged lowpriority cell gets served given that the tagged low priority cell will get served, is:

TIle mean waiting time of the low priority cell given that it gets served, WL , is:

K-l K-l-i

Po + L L Pi,jUi+j+li=O j=O

17

5 Approximate analysis of the switch

In this section, we describe a simple approximation algorithm for the analysis of the ATMswitch. The approximation algorithm utilizes expression( 3) for a , and the exact resultsobtained in sections 3 and 4. The algorithm is an iterative scheme. III each iteration weneed the cell loss probability for each packet processor queue irrespective of the priority ofthe lost cell. For efficiency purposes, we calculate this cell loss probability using an algorithmfor analyzing a single-class IBP /Geo/l/K+l queue that was presented in [27].

The iterative approximation algorithm involves the following steps:

step o. Set initial value of the utilization of the ith cell processor, p~O), i := 1,2· .. N.

step 1. Compute the probability of retransmission of a cell due to contention ill the ClosATM switch, 00(0), using expression 3.

step 2. Solve the IBP/Geo/1/K + 1 queue with a single class of cells using the solutionmethod in [27] in order to find the probability of cell loss due to the fact that the buffer

is full, IK.

step 3. Compute the utilization of the cell processor, pP) :

pP) = a(I - q) (I-,K)2-p-q

,i=1,2.··N.

step 4. Use expression 3 to compute u(1) the probability of retransmission of a cell due tocontention in the Clos ATM switch with the new cell processor utilization, p~l), i =1,···,N.

step 5. If lu(l) - u(O)1 < e then o = u(1) and go to step 6, else set u(O) = u(1) and go to step

2.

step 6. Solve the two-class I BP/Geo/I/ K + 1 queue with head-of-line with no preemptionand push-out to find the blocking probability of the high priority cell, PK,O'

step 7. Compute the probability that a low priority cell gets served and its mean waiting

time.

18

6 Numerical Results

TIle approximation algorithm described in the previous section was employed to analyze a16 X 16 bufferless Clos ATM switch. Each switching element was a 4 X 4 crossbar switch.The system size of each cell processor queue was set equal to 32. The arrival process toeach cell processor was assumed to be an IBP with a == 1. That is, during the busy periodeach slot contains a cell. The approximation results were compared against results obtainedby simulation. TIle results are summarized in figures 7 to 19. Figures 7 to 11 are for asymmetric case, and figures 12 to 19 are for two different asymmetric cases.

In the symmetric case, we assume the same arrival process to each input link withan average arrival rate 0.3. Each arriving cell chooses one of the destination output linksuniformly. In figures 7 and 8, we plot the approximate and simulation results for the meanwaiting time for high respectively low priority cells as a function of (3 for various 0 2 values.In figure 9, we plot the approximate and simulation results for the probability that a lowpriority cell gets served given that it entered the queue as a function of (3 for various C 2

values. The approximate and simulation results for tile cell loss probability for low prioritycells is given in figure 10 as a function of C 2 for (3 equal to 0.3. We note that the cell lossprobability of the low priority cells comprises the probability a low priority cell will arriveto a full queue and the push-out probability. In Figure 11, we plot the approximate andsimulation results for mea.n waiting time for high and low priority cells as a function of C 2

for (3 equal to 0.5.

In the results given in figures 7 to 10, the minimum relative error was 1%, themaximum relative error was 15%, and the average relative error was 7%. Most of theapproximate results had a relative error less than 5%. The curves given in figures 7 and8 show how the mean waiting time is affected by 0 2 and {3. When C2 is high (i.e. burstyarrivals), the mean waiting time of the low priority cells drops when f3 == 0.9. This is dueto the fact that most of the low priority cells are pushed-out of the buffer by high prioritycells. Figure 9 shows how the probability that a low priority cell gets served is affected bythe value of C 2 and (3. In particular, we note that this probability significantly drops as C2

Increases.

In the two asymmetric cases considered here, the following assumptions were rnaderegarding the arrival processes.

case 1: Ai == 0.4,i == 1,2,···,8 with C2 == 100,Ai == 0.1, i == 9,10, · · · ,16 with C 2 == 10.

case 2: Al == 0.4, Al6 == 0.1 , and the values of the remaining Ai'S were linearly distributedbetween 0.4 and 0.1. TIle value of 0 2 was the same for all 16 arrival processes and itwas varied from 1 to 500.

19

Mean Waiting Time oftbe High Priority CeU

20

----- C··2dO, Appro:dms.tlon

--0-- C··2=50, Simulation

---- C......2=10, Approximation

--.-- C**2=10, SlnmJation

---- 0·2=100, Approximation

---.-- <:-"'1=100, Simulation

1.00.80.60.40.2

5

-m-- <:**2=1,Approximation

-+-- <:*·2=1. SimulationO-t-""'T""""'"'~--r---r----,-~--r--r---r---"--,.--,,......--,r--_

0.0

10

15

Figure 7: Mean waiting time of the high priority cell; symmetric case.

In both asymmetric cases, it was assumed that each arriving cell chooses one of the destina­tion output lines uniformly.

Figures 12 to 14 are for the asymmetric case 1, and the remaining figures a.refor the asymmetric case 2. In all the figures we give approximate and simulation results forthe most utilized input queue and the least utilized input queue. The relative error of theapproximate results is similar to the one observed in the symmetric case discussed above.Finally, we note that the behavior of the performance curves given in these figures is similarto the one observed in the symmetric case.

7 Conclusion

In this paper we presented a queueing model of a blocking bufferless synchronous ClosATM switch with input buffers. The arrival process to each input port was assumed tobe bursty and it was modelled by an Interrupted Bernoulli Process. Two classes of cellswere considered. The service in each input buffer was head-of-line without preemption. Inaddition, it was assumed that a high priority cell arriving to a full queue can displace alow priority cell( space preemption). An exact analysis of the input buffer was presented.The results obtained were used in an approximation algorithm for the analysis of the ATIvIswitch. Validation tests using simulation data showed that the approximation algorithm has

20

Mean Waiting Time ofthe Low Priority Cell

100

80

60

40

20

• • •

--A- C.·2=IOO, ApproxlrTUltion-....- C"Z=lOO, Slmulatl~n

_____ C••Z=50, ApproxJmation

--0-- C··Z=50, Simulation

--a-- C••Z=10, Approximation~ C.·2=10, Simulation

____ e*·1=1, Approdmatlon

--.-- C••Z=l, Simulation

1.00.80.60.40.2o-l--.-JII::::;:~==;:::::II:=;=~~:R:;==tIl=::;::::II~=*:::;:a..,--.,

0.0

Figure 8: Mean waiting time of the low priority cell; symmetric case.

~ C"=l, Apprmtmatlon--...- C··Z=l, Simulation---- C··Z=10, Approxfmntlon

-.-- C··Z=10, Simulation

---- C··Z=-<G, Approximation-0-- C··Z=50, Simulation

--.-- e*·Z=l00, Approximation

----6-- e*·Z=100, Simulation

--It-' C**Z=ZOO, Appro~matlon

---+-- C··Z::200, Simulation

0.8 1.0

P0.60.40.2

1.0

0.8

0.6

0.4

0.2 -+---T-r--r----,r---r-~-or__r_.....,.-.,.___r_-.,..___r____r~

0.0

Probability that a lowpriority cell gets served

Figure 9: Probability that a low priority cell gets served; symmetric case.

21

Cell Loss ProbabUltyforLow PrIority Cells

10 0

--m-- Approximation-..- Simulation

60040030020010010 - 5-t-----r---~....,.--r---'r----r----­

o

Figure 10: Cell loss probability for low priority cells; symmetric case, (3 == 0.3.

Mean Waiting TIme

100

Low Priority Cell:

~ ApproximAtion

~ Simulation

p=o.5

High Priority Cell:

_____ Approximation

--+-- Simulation

600

Figure 11: Mean waiting time for low and high priority cells; symmetric case, {3 = 0.5.

22

ProbabWty that a lowpriority cell gets served

Low UtlllzatlOD Input Queue:

1.0 -+-- Slmal8tlon

--a-- Approximation

0.8

0.6

Hlgh Utillzatlon Input Queue:

-+-- Simulation

-'It-- Approximation

0.4

1.00.80.60.40.2

0.2 -J- -~--r---y-___r-r___r__~___r___,r__~~__,

0.0

Figure 12: Probability that a low priority cell gets served; asymmetric case 1.

MeaD Waiting Time of theHIgh Utillzed Input Queue

120

----- Approximation

-...- Simalation

HIgh PrIority CeO:

Low Priority Cell:

1.0

-a-- Approximation

-+-- Simulation

0.80.60.40.2

o.+-_a::;::::::::tI~::;:::II~=*==;:=-=;::=*=:::;:~- _0.0

60

20

40

80

100

Figure 13: Mean waiting time for the low and high priority cells of the high utilized inputqueue; asymmetric case 1.

23

Mean Waiting Time of theLow Utilized Input Queue

6 Low Priority Cell:

5

4

----.- Simulation

--a- Approximation

3

2

HIgh Priority Cell:

-.-- SlmulnUon

~ Approximation

1.00.80.60.40.2O+-...,....--....~-..-~-r---.--..............-....--_......._-....-

0.0

Figure 14: Mean waiting time for the low and high priority cells of the low utilized inputqueue as a function of {3; asymmetric case 1.

Mean Waiting Time of theHlgb Utllized Input Queue

120

100

80

60

40

20

Low Priority Cell:

-a--- Approximation

--0-- Simulation

High Priority Cell:

--m-- Approximation

-+-- Simulation

1.00.80.60.40.2

oL.---!R::;::~==;::R::::;:::::~~*=;:=tlIt::::~~----r-~--r---,

0.0

Figure 15: Mean waiting time for the low and high priority cells of the high utilized input

queue as a function of (3; asymmetric case 2.

24

High Priority CeU:

Low Priority CeU:

--+-- Approximation

~ Simulation

-----D- Approximation

-+-- Simulatlqn

20

40

60

Mean Waiting Time of theLow UtUlzed Input Queue

80

1.00.80.60.40.2o -l-~t::;=:=-=:::::;::::I1::;:~:::;::a:;~~~....--....-....--...--..

0.0

Figure 16: Mean waiting time for the low and high priority cells of the low utilized inputqueue as a function of (3; Asymmetric case 2.

Mean Waiting Time of theHigh UtUlzed Input Queue

50

40

Low Priority Cell:

--.- Approximation

---...- Simulation

30

20

10

HIgh Priority Cell:

------ Approximation

--+-- Simulation

600500400300200100

o~::::$:~~:;::::~==;:=:;::=::::;::::=;:=;::=:::e-__o

Figure 17: Mean waiting time for the low and high priority cells of the high utilized inputqueue as a function of C2 with (3 = 0.3; Asymmetric case 2.

25

Mean Waiting Time of theLow Utilized Input Queue

50

Low Priority Cell:

--...- Simulation

-'It- Approximation

High Priority Cell:

--...- SImulation

~ Appromn.tton

Figure 18: Mean waiting time for the low and high priority cells of the low utilized inputqueue as a function of 0 2 with {3 = 0.3; Asymmetric case 2.

Cell Loss Probability ofthe Low Priority Cell

10 0

600500

C··2

400

-iii'- Approximation

--+-- SImulation

300

Low Utillzed Input Queue:

--0-- ApproxJm.tlon

---.--- 51ma.etlon

High Utilized Input Queue:

200100

10· 6+-J..L...,.-..,-----r--r----r--r-__,._-~--,.--.._--r-___,

o

Figure 19: Cell loss probability of the low priority cell; asymmetric case 2.

26

a good accuracy. Using the analytic techniques developed in this paper, various in terestingperformance measures such as cell loss probability, mean waiting time, probability that a

low priority cell gets served, were obtained.

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29