Delay analysis of a modified cut-throughswitching for multipacket messages

M. llyas, M.Sc. Ph.D., Mem.I.E.E.E., and H.T. Mouftah. M.Sc. Ph.D..Mem.I.E.E.E.

Indexing terms: Switches and switching theory, Computer networks, Resource sharing. Reassembly delay.Hybrid switching

Abstract: A new switching technique called 'virtual cut-through' has been recently proposed for computercommunication networks. The paper presents the analysis of 'virtual cut-through' for multipacket messages.Also introduced is a modified virtual cut-through switching technique that allows partial cuts. The performanceanalysis of partial cuts is presented. Partial cuts have been shown to be more advantageous from the practicalpoint of view. It has been observed, from the analytical results in the paper, that the use of partial cuts gives anassisting hand to boost the performance of a network exactly when it is badly needed, i.e. when the value of lineutilisation lies between 0.4 and 0.8. Various performance curves that provide deep insight and better under-standing of the behaviour of this switching technique are also given.

1 Introduction

The importance of remote processing is growing at anenormous rate. Computer networks are used to accom-plish the task of remote processing. Performance and effi-ciency of a computer network depend on many factors(buffer size and network topology etc.). Performance cri-teria include the network's response time and throughput.Response time is the time taken by a message to travelfrom its source to its destination, and throughput isdefined as the number of messages successfully transferredper unit time. These two factors are intimately related toeach other. When response time of an arbitration networkis reduced, the throughput of that network receives aboost. Efficient switching techniques guarantee the lowervalue of response time and higher throughput. There arethree basic switching techniques used in computer commu-nication networks:

(i) circuit switching(ii) message switching

(iii) packet switching.

Efforts have been recently directed towards designinghybrid switching techniques, i.e. combinations of morethan one switching technique to obtain better network per-formance. The latest innovation in this direction was pro-posed by Kermani [4] and was named 'virtualcut-through' switching. In this technique a message is notstored fully at an intermediate node if the node is free. Theheader of the message, however, has to be processed inorder to choose the next outgoing line for that particularmessage. It means that the message can start its onwardjourney through the network immediately after its headerhas been processed. When this happens, the message issaid to have made a full cut. This, however, happens onlywhen the intermediate node is found empty by an arrivingmessage. The results presented in Reference 4 deal withsingle packet messages and full cuts only. This paper pre-sents the analysis of 'virtual cut-through' switching formultipacket messages and also introduces a modifiedvirtual cut-through switching technique that allows partialcuts. The idea of a partial cut is motivated by the fact that,when an arriving message finds the node busy servicing theonly packet present at that node, the newly arrived packet

Paper 3573E {C3), first received 12th July 1982 and in revised form I2th September1984

Dr. llyas is with the College of Engineering, Department of Electrical Engineering,Florida Atlantic University, Boca Raton, Florida, USA. Dr. Mouftah is with theDepartment of Electrical Engineering, Queen's University, Kingston, Canada K7L3N6

does not have to be fully stored at that particular nodeand can start its onward journey as soon as the precedingmessage's service is finished. Gain in terms of reduction, i.e.the delay per message for a partial cut, is, of course, lessthan that in the case of a full-cut, but the results haveshown that the use of partial cuts gives a boost to the per-formance of a network when it is urgently needed, i.e.when line utilisation is in the range 0.4 to 0.8. This aspectof partial cuts indicates its practical versatility. The modelused for the purpose of carrying out the analysis is dis-cussed in Section 2. Mathematical formulation of theproblem and relevant assumptions are discussed in Section3. Section 4 is devoted to the discussion of numericalresults. The results presented in this paper show the useful-ness of cut-through switching with partial cuts.

2 Model

The network is assumed to have N nodes and M channels.The original messages entering the network come from anexponential distribution as far as their lengths are con-cerned. The average length of these original messages isl//z bits per message. Their arrivals obey the Poissonprocess with an average of X messages per second. Onentry to the network, the messages are segmented intopackets of maximum length lp bits. Thus the arrival rate (interms of packets) increases, and average length decreases.However, the product of arrival rate and average lengthremains the same [I ] . If the total traffic entering thenetwork in terms of packets per second is denoted by y,then this quantity is given by

N N

y = I 1 (i)

where yu is the average number of packets entering thenetwork per second from source i to destination j . N is thetotal number of nodes. Similarly the total traffic travellingin the network channels Xp(t) packets per second is givenby

= t x (2)

where M is the total number of network channels and Xpi

is the traffic in packets per second entering the ith channel.It can be immediately seen that Xpi, and thus X{

p\ dependon the routing algorithm of the network. It can also beseen that X(

p] is always greater than, or equal to, y. The

equality occurs when there are direct links between all host

IEE PROCEEDINGS, Vol. 132, Pt. E. No. 1, JANUARY 1985 45

and destination nodes. The average number of store-and-forward nodes traversed by a packet is simply given by

n = (3)

Full cuts or partial cuts can only be performed at h — 1nodes. An average number of store-and-forward nodesbeing in hand, one can build a simplified network modelfor mathematical tractibility, as is shown in Fig. 1. This

destinationA, | * 2

q, Q2

Fig. 1 Analytical model ofM/G/1 queues in tandem

k, = ks + kel i = 1* f = ^» i + ^ i - i ( l "~ flf-i) i = 2 , 3 , . . . , n

model is a general model and facilitates use of the indepen-dence assumption [5]. This assumption makes the mathe-matical analysis extremely tractible.

3 Mathematical formulation

The original messages entering the network have anaverage message of l/p. bits and follow exponential dis-tribution. Upon entry the messages are segmented intopackets of maximum length lp. After this segmentationprocess the packet lengths no longer follow exponentialdistribution and the average packet length xp turns out tobe

(4)

and the average number of packets per message is given by1/[1 — e~ftlp']. The line utilisation p, however, remains thesame in both cases [1]. As the packet length distribution isno longer exponential, the results for the M/M/l queueingmodel cannot be used. For the purpose of this analysis theproblem will be formulated as M/G/l queue. To find outthe probability of performing full cut or partial cut, theprobability of the system being empty and the probabilityof having one packet in the system are needed, respec-tively. These probabilities are calculated with the help ofthe probability generating function P{z).

After segmenting the original messages into packets ofmaximum length lp the probability density function forpacket lengths is given as follows:

/* =

fie

0

for 0 < x < lf

X — lp

otherwise

(5)

The Laplace transform of the probability density functionF(s) is found to be

F(s) =+ s

M _ (6)

The moment generating function A(z) could be obtainedby replacing s by Xp — Xp z in F(s), which gives

A(z) =f -z)

i (7)

As a check, when the value of z is put as unity, then thewhole expression should be equal to one, and in the aboveexpression A(l) = 1. Now the probability generating func-tion P(z) is given by

P(z) =- p)(z - \)A{z)

z - A(z)(8)

where p is the line utilisation and is given by

P = V c (9)

where tc = xJC: C being the capacity of the link in bitsper second and xc is the average packet length. (For sim-plicity the capacity of each link will be assumed to be thesame, and the value of qt is such that Xpl = Xp2 = • • • =Xpn = Xp). Now the probabilities of system occupancy aregiven by

dkp(z)

dkz

k\(10)

z = 0

where pk is the probability of having k packets in thebuffer. From here Po and Px are obtained to be used as theprobability of making a full cut or partial cut, respectively.(Recall that the independence assumption is being used.)

The average number of full cuts per packet Nf is givenby [4]

Nf = P0(h - (11)

By the same argument it can be shown that the averagenumber Np of partial cuts is given by

Np = P^h - 1) (12)

Reduction in delay per packet Sf due to a full cut at anode can be shown to be

= tc - t (13)

where t0 is the transmission time of the header and is equalto xo/C: X0 being the header length in bits. It is assumedthat the processing delay for the header is negligible, andthe outgoing channel can be decided immediately after theheader has been received. Total reduction in delay perpacket due to full cuts is, therefore, equal to NfSj. Forcalculating the reduction in delay due to partial cuts, theservice time remaining to complete the service of thepacket already in the service facility is needed. This quan-tity can be shown to be [2, 6, 8]:

Xptf/2 (14)

where t2c is the second moment of the service times.

Reduction in delay in the case of a partial cut per packetwill, therefore, be

[•¥-] (15)

The total reduction in delay per packet due to partial cutsis, therefore, equal to Np Sp. If the total delay per packetwithout performing any cuts is denoted by T(^c and thatwith cuts by Tj^, then, for single packet messages (i.e. mes-sages with length less than lp),

TJ2 = T{^oc - total reduction (16a)

so

X(S) = T(5) — JV S — N S (166)

For the case of multipacket messages, however, the formi-dable problem of reassembly has also to be considered.

46 IEE PROCEEDINGS, Vol. 132, Pt. E, No. 1, JANUARY 1985

For calculating the reassembly delay, an approximation[1], which expresses the interpacket gap as a function ofline utilisation and service time, will be used. The inter-packet gap g between two packets belonging to the samemessage when they travel through n nodes is given by

9 =1 - p

(17)

Now if a message has been segmented into m packets, thetotal gap between its first and last packet will be

1 - p(18)

If the value of p was very low, all the packets of a messagewould have reached the destination one after another. Butat a significant value of p this gap will exist and has to beadded to the delay of the first packet. However, when cutsare made, the effective value of the average number ofnodes decreases, depending on the value of p, and this willcause a reduction in the interpacket gap. Effective numberof nodes neff is given by

«e// = (^o + Pi)(n - 1) + 1 (19)

Effective interpacket gap is, therefore, given by

9eff =(1 _

\-p(20)

(21)

Reduction in reassembly delay Sr will thus be given by

Sr = (m- 1X0 - geff)

In a multipacket message the delay for the first packet isthe same as any other single packet message. The rest ofthe packets follow after an interval equal to the interpacketgaps. The total reduction in delay R for a multipacketmessage having m packets will be given by

R = NfDf + NpSp + Sr (22)

So for a multipacket message case the delay with cuts isgiven by

- R (23)

where T^l and T{^] represent the delay for multipacketmessage without and with cuts, respectively. Here T^J. isgiven by

<m -

and T{£]oc for single packet messages is given by

(24)

(25)

Once the delays for single packet messages, T ^ , and thosefor multipacket messages, T\%\ have been obtained, thenoverall average delay Twc per message can be obtained asfollows:

TWC = a - (26)

where /? is the probability that a message has a length lessthan lp. The corresponding overall average delay Twoc permessage without performing any cuts can also be similarlyobtained.

In the above analysis, the main assumption used is theindependence assumption due to Kleinrock [5, 1]. Thisassumption gives good results when topology of the arbi-tration network is sufficiently complex. To incorporate

that feature in the model discussed here, lateral entry oftraffic has been considered at every node. This leads tostraightforward mathematical analysis. The second majorassumption used in this paper is related to the output ofthe service facility. The assumption made here is that theoutput of a queue which makes a part of input to the nextqueue follows the Poisson process. This assumption isclosely related to the first one and works well when thenetwork is complex. More on this assumption can befound in References 5, 7 and 8. Apart from this, propaga-tion delays and header processing delays were assumed tobe negligible. None of the above assumptions is severe, andso the results obtained in this work are reasonably accu-rate.

4 Numerical results

A variety of results have been obtained, to improve under-standing of the effects produced by partial cuts. It has beenobserved that, at the extreme values of p, i.e. 0 and 1, thereduction in delay due to partial cuts decreases to zero. Apartial cut, however, reduces the delay by more than onethird of that achieved by a full cut, at some intermediatevalue of p. This means in practice that partial cuts arehelpful in reducing the average delay per message at thetime of need.

Fig. 2 shows the contribution in reducing the averagedelay per message due to full cuts and partial cuts against

0.2

0.1

" 0.01o

0.0010.2 0.4 0.6 0.8 1.0

Fig. 2 Reduction in delay against line utilisation

- =2000 bits/*/„ = 1200 bitsxo = 50 bitsc = 50 Kbit/sh = 7A =0.5

line utilisation. It can be seen that the contribution due tofull cuts reduces as line utilisation is increased and finallyapproaches zero at p = 1. The contribution due to partialcuts is zero at the lowest value of line utilisation; itincreases as the value of p is increased, reaches its peakvalue at about p = 0.6 and then starts falling, meeting zeroat p = 1.

Fig. 3 shows a family of curves depicting delays againstline utilisation. The delays increase monotonically with theincrease in line utilisation. But the delays obtained by

IEE PROCEEDINGS, Vol. 132, Pt. E, No. 1, JANUARY 1985 47

using partial cuts and full cuts together are shown to bealways less than with no cuts. These results also vindicatethe versatility of partial cuts.

In Fig. 4 the ratio Twc/Twoc is plotted against line uti-lisation of the network. Different curves correspond to thedifferent number of hops. As the value of line utilisationincreases, all the curves tend to converge to one point. Thisis understandable because, at higher values of line uti-lisation, full cuts or partial cuts do not help much, and thetwo delays (with cuts and without cuts) do not differ much.Their values become equal at p = 1.

In Fig. 5 the ratio Twc/Twoc is plotted against packetlength lp for different values of line utilisation. These curvesshow that for each value of line utilisation there is anoptimal value of lp which minimises the ratio Twc/Twoc. Thelocus of this optimal packet length is also shown. This pro-perty could be used to design a flow control for a networkon the basis of packet length. The packet length can be

i.o-

0.8

0.6

0.4

0.2

full and partial cuts,full cutSvno cuts.

0 0.2 0.6 0.8 1.0

Fig. 3 Average delay per message against line utilisation1 -,™ . c =50 Kbit/s

= 2000 b ts - .,H n =1lp = 1000 bits A = 0.5.v0 = 50 bits

1.0

08

0.6

0 4

0.2

0.2 0.4 0.6P

0.8 1.0

Fig. 4 Ratio of delay with cuts to the delay without cuts against lineutilisation

= 2000 bits

/„ = 1000 bits

xo = 50 bitsc = 50 Kbit/sk =0.5

reduced at higher values of line utilisation and vice versato have optimal performance.

5 ConclusionsAn extremely useful improvement is proposed in 'virtualcut-through' switching, along with its extension to multi-packet message case. In practice the improvement, which isthe result of using partial cuts in addition to full cuts, isconsiderable. The use of partial cuts in cut-through switch-ing does not affect the delay performance at the extremevalues of line utilisation, but brings the delay to a lowervalue for medium-range values of line utilisation. Also,apart from the efficient delay performance, partial cuts arenot difficult to implement. All these advantages of the cut-through switching with partial cuts should make it accept-able for implementation.

Several numerical results have also been presented forthe delays without using cuts, with full cuts and with bothfull and partial cuts. These results depict the behaviour ofpartial cuts against different network parameters.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

p=0.3

= 0.2

0.1 K IK 10K

Fig. 5 Ratio of delay with cuts to the delay without cuts against packetlength

I c = 50 Kbit/s- = 2000 bits - _ 6

.v0 = 50 bits '*• = ° - 5

6 References1 COLE, G.D.: 'Computer network measurements: techniques and

experiments', Computer Science Department, UCLA, Report UCLA-ENG-7165, 1971

2 COX, D.R.: 'Renewal theory' (John Wiley, New York, 1962)3 INOSE, H., and SAITO, T.: 'Theoretical aspects in the analysis and

synthesis of packet communication networks', Proc. IEEE, 1978, 66,(11), pp. 1409-1422

4 KERMANI, P.: 'Switching and flow control techniques in computercommunication networks', Computer Science Department, UCLA,Report UCLA-ENG-7802, 1978

5 KLEINROCK, L.: 'Communication nets' (McGraw-Hill, New York,1964)

6 KLEINROCK, L.: 'Queueing systems. Vol. II—Computer applica-tions' (John Wiley, New York, 1976)

7 MIYAHARA, H., TESHIGAWARA, Y., and HASEGAWA, T.: 'Delayand throughput evaluation of switching methods in computer net-works', IEEE Trans., 1978, COM-26, pp. 337-344

8 SCHWARTZ, M.: 'Computer communication network design andanalysis' (Prentice-Hall, New Jersey, 1977)

9 KERMANI, P., and KLEINROCK, L.: 'Virtual cut-through: a newcomputer communication switching technique', Comput. Networks,1979, 3, pp. 267-286

48 IEE PROCEEDINGS, Vol. 132, Pt. E, No. I, JANUARY 1985