Message Path Delays in Packet-Switching ... - Google Sites

186

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. C O M - ~ ~NO. ,

2,

FEBRUARY

1975

in Packet-Switching Communication Networks

MessagePathDelays

IZHAK RUBIN

Abstract-A communicationpath (in isolation) in a packet-switching store-and-forwardcommunicationnetwork, such .as a computer- or satellite-communicationnetwork, is considered. Messages are assumed to arrive according to a Poisson stream, and messagelengths are considered to be random variables governed byan arbitrary distribution. Message lengths are divided into fixed-length packetswhichare sent independentlyover the N-channel communication path in a store-aid-forward mamier, and are reassembled at the destination terminal. Expressions for the distributions of the message waiting and delay tiines over the path are derived. Also, weobtain the limiting.average message waiting times and required bufFer sizes at the individual channels. The overall message waiting time is observed to depend only on the minimal channel capacity. Thecase of exponentiallydistributed message lengths serves as an illustrating example.

tionnetwork. Assuming Poissonarrivals at the source station of this path andrandom message lengths governed by an arbitrary aistribution, we derive exact expressions for the distribution of the message delay over the communication path. We also obtain steady-state expressions for the average message waiting times and required buffer sizes at the indiiidualchannels. Exact expressions for waiting and delay time ldistributions’, as well as busy-period characteristics, along a communication path in a store-and-f,orward communication network, have recentlybeen derived in [1] and [2]. These papers assume k e d message lengths. The results obtained in this paper thus follow from those obtained in [1] and [2]. Approximate time-delay resultsfor store-and-forward INTRODUCTION I. communication networks, utilizing an “independence assumption’’ (which requires rechoosing the message COMMUNICATION networkisrepresented asa weighted graph. The branches of the graph represent length, at random, at any station) andassuming exponenthe communication channels, while the vertices represent tially distributed message lengths, are reportedin [3] and (source, repeater,,or destination type) stations with stor- [4] and the references therein. Many of the recent timeage (queueing) facilities.The branches areassigned capac- delayreportshave been associatedwith the Advanced ity weights. Messages arrivea t random at a sourcestation Research Projects Agency (ARPAj computer-communiand follow a specific route in the network towards their cation network (see [4]). The latter employs packets of destination station.Message lengths are usually considered 10Ob-bit length (and some others of shorter length, for to be random variables.I n a packet-switchingcommunica- situations in which the packet is not filled). We note that our delay analysis involves a single path tion network, the message is divided at thesource station into fixed-lengthsubmessages, called packets. Those pack-in isolation, so that themessages in this pathdo not intermesetsarethensentindependentlythrough the network fere with and are not interfered with by any other towards the destination station. A t the latter stations,all sages in the network. Messages arriving at the path are the packets associated with a specific message are reas- spaced out in contiguous packets by the first channel. If the timing of sembled. The wholemessage is then transferred to the the nextchannelhasahighercapacity, packets is preservedbut the durationof each is shortened. destination terminal (being the destination computer in case of a computer-communication network, or a specific If the next channel has a lower capacity, the packets will terminal in a satellite-communication network). The net- be extended and if messages are close enough together, further delayswill ensue. We show that thedistribution of work is also assumed to operate in a store-and-forward manner, so that at each station a queue of messages is the overall message waiting time in the path is equal to generated and served accordingto a first-come first-served that obtained by presenting allthe messages to thechannel with lowest capacity only.discipline. Preliminary notions and definitions are introduced in In this paper,we are considering an arbitraryN-channel path in apacket-switching store-and-forwardcommunica- Section 11. Message time delaysover a single-channel path are obtained in Section 1II:In Section IV, message waiting and delay times over an N-channel communication path Paperapproved bythe AssociateEditor forComputerCommunication of the IEEE Communications Society for publication are derived. As an illustrating example, we consider in after presentation at the 1974 IEEE International Symposium on Section V the case of exponentially distributed message Information Theory, Notre Dame, Ind., October 1974. Manuscript the traffic received February 21, 1974; revised August 21, 1974. This work was lengths, and plotthe resulting time deiay versus supportedinpart bythe Officeof Naval ResearchunderGrant intensity for several values of average packet to message N00014-69-A-0400-4041, and inpartby the AdvancedResearch lengths. The special case of packets of vanishing length, Projects Agency under Grant DAHC15-73-C-0368. The author is with the Department of System Science, School of andarbitrary message-length distribution,is then obEngineeringandAppliedScience,University of California, Los served. Angeles, Calif. ,

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187

RUBIN: MESSAGE PATH DELAYS

11. PRELIMINARIES A communication path, asshown in Fig. 1,is considered. The ith channel, whose capacity is Ci [bits/s], is represented by the branch (ui,u;+l) between vertices ut and v,+~, i = 1,2, , N . The vertices represent stations or buffer systems equipped with (infinite space) memory storage units or queueing facilities. Messages arrive at input station v1 a t random times, following a Poisson stream with arrival intensity X [messages/~]. The message length is assumed to be random. We let Y,[bits/s] denote the length of the nth arriving message at ul, and assume { Yn,n 2 1) to be an independent identically distributed (iid) sequence of (nonnegative valued) random variables, governed by the distribution function F y ( y) , where

2 '

1'

MESSAGE PACKETS ARRIVALS GENERATOR I 'PG

2'

*--'N-1'3

N '

MESSAGE

' N - ' L \ W

N ' -I

PACKETS REASSEMBLY

- .-

F y ( y ) = P { Y n 5 Y).

(1)

The average message length is p-l [bits/message], p-1 =

E ( Y,)

=

/

m

ydFy(?l).

(2)

0

At ul, before being processed through the communication path, the message is divided into (an integral number of) packets of fixed length. We set the packet length to be a[bits/packet], and denote the number of packets representing the nth message by dl,. Thus, (n/l,,n 2 1) is a sequence of iid random variables following the probability measure g(m), where g(m)L P ( M , = m )

=

F ~ ( ~-~F z~ )( ~ [ v zi]), nt 2 1.

(3)

Clearly, Cz-lg(m) = 1 and the average number of packets per message is denoted as

DEPARTURES

2'

Fig. 1. (a) Communication path. (b) Path i n a packet-switching communication network.

through the ith channel if it is free, while, if the channel is busy, the packet joins the queue a t ui, and is served on a first-come first-served basis. Whilepackets are processed independently at any station vi, i = 1 , s . . , N , at the destinationterminal UN+1 all the packets belonging to one message are assembled and subsequently leave the network (being now transferred to the destination computer). Each channel with its storage facilities can be considered as queueing system. For that purpose, we consider a packet which is to be transmitted over the ith channel to be a customer which requires service from the ith server. A message is thus considered to be a group of customers. Customers thus arrive at u1 in groups, where the arrival process is Poisson with rate X, and the group size is governed by distribution g ( m ). The service time required by a customer (packet) from the server (channel) i is equal to his transmission time over the channel and is given by ai = a/Ci

[s/packet].

(6)

The following notations will be utilized throughout the paper. We let X t t i ) denote the number of packets stored = mg(m) [packets/message]. (4) a t ui or being transmitted through channel i a t time t . n=l Thus, { Xt(Q,t2 0 ) is the queueing processassociated with For example, if the message length is exponentially dischannel i. We assume = 0,i = 1,. , N . The (rantributed with mean p-l, dom) instants of arrival of packets at ui are denoted by = 1 , 2 , - - . ; j , = 1,2,...,M,), where t,Ji) is the FY(Y) = (1 - exp C-MYl)U(Y) (5a) {t,,in(i),n instant of arrival at ui of the jth packet associated with where u (y) denotes the unit step €unction; then ( 3 ) yields the ,nth message. The instants of message arrivals at vi the probability y,(m) , where areset to be {&(i),?z = 1,2,. ) , where Zn(i) g tn.l(i). Similarly, we denote the instants of packet and message ye(nz) = p p - 1 , 'm 2 1 (.5b) departure from channel i by { r , J i ) ) and { P , t i ) ) , respecand tively, where Pn(i) and r,Ji) is thedeparture time from channel i of the jth packet associated with the nth Q = exp [-Pal, P = (1 - d , (jc) message. Clearly, since we have batch arrivals at u1, we so that the number of packets per message follows a geo- have tn,j(l) = tn,l(l)= for each j,n. The packets assometric distribution. Clearly, if the message length is not ciated with a specific message are then ordered according larger than a packetlength, we have g ( l ) = 1 and to their order of service. Thus, the h-th packet of the nth M , = 1 with probability 1 (and the situation studied in message arriving at v1 is the kth one to be transmitted over [l J subsequently follows). channel 1, among the message M , packets, and it departs The packets derived a t u1 from each message are then into u2 a t time rll.,k(l).Clearly, for a conm~unication path, transferred through the communication path in a store- tn,j(i+l) = r n.3. ( i ) tn (i+l) = $nco,j = 1,2,. .,N - 1. and-forward (packet-switching) manner.Thus, a packet The waiting time a t v1 of the jthpacket associated with arriving a t ui, i = 1,2, . , N , is immediately transmitted the nth arriving message is denoted as W,Ji). The nth

a

m

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188

IEEE TRANSACTIONS ON COMMUNICATIONS, FEBRUARY

message waiting time a t

vi,

F,(i)is set as

Wn" -. A W,,I(~), %) I

(7)

and is thus equal i o the waiting timeof the first associated packet. The packet delay time over the ith channel -y,+$) is given by the total of its waiting time and transmission delay time a t channel i. Thus, we have -y?,j( 0,

(13)

where

(8)

For the nth message, we define its delay time over the ith channel, i = 1,2, , N , by

--

Tn(i) = y n , M n ( i ) - t,,l(i)

[x] is the largest integer not larger than x, and [ F ( t ) Y * denotes the nthconvolution of F ( t ) . The limiting message mean waiting time is given by

(9)

as the time difference between the departure of the last associated packet and the arrival of the message at vi. Similarly, the overalltime-delay of thenth message through the N-channel communicationpath is given by

7,

= r n , M n ( N ) - tn ,I ( I ) ,

where

( 10)

as the time-difference between the instant the last associated packet leaves the Nth channel and the instant of arrival of the nth message a t vl. is the coefficient of variationassociatedwithrandom In this paper, we will derive the steady-state distribuvariable M, and distribution g ( m ) . For pl 2 1, we have tions for the overall message~time delays and the average W(I)( t ) = 0, for each t > 0. message waitingtimesanddelays along the individual Proof: The theorem follows from (11) since for a channels. M/G/1 system we have Wn+l = [W, X , - T,+1]+, where the service time X , and the interarrival time T n f l 111. THE SINGLE-CHANNEL PATH are independent random variables, the latter being expoConsider the case N = 1, so that the pathincludes only nentiallydistributed. Hence, (13) and (14) follow (see a single channel. In this case, if we are interested in the [5, pp. 255-2561). Equation (14) is known as the Polpackets' waiting-times, we are considering a MIDI1 laezek-Khintchine Q.E.D. formula. queueing system with group arrivals (see [5]-[7]). For The message delay timeT n ( l )defined by (9) is now equal' this system,customers(packets)arriveaccording to a to the overall message delay time -fn of (10). The latter Poisson stream with intensity X , and each requires a fixed are clearly given by service of length ul. Waiting time resultsfor the individual packets readily follow. However, we are interested herein the message time-delays. For that purpose, we need to Hence, we conclude the following results, observing @,(I) obtainthe distribution of the message waitingtime and M, to be statistically independent. @,(I) = W,,l(l). The latter random variable satisfies the Corollary 1: For p1 < 1, and a single channel path, the relationship steady-statedistribution of the message delaytime is F,+l(l) = [ W n ( l ) Mnul - Pn+l(l)]+, (11) given by

+

+

m where [X]+ p max ( 0 , X ), and pn+l(i) = ln+l(i)de- y ( I ) ( t ) p lim P{T, _< t ] = W(l)(t- maI)g(m), (17) notes the ( n 1)st message interarrival time at channel n- m Wl=l i. Relation ( 11) indicates that for deriving the message waiting times,we need consider a M/G/l queueing system where W(I)( t ) is given by (13). The limiting mean delay with unit Poisson arrivals of intensity X and service times time is equal to equal t o M,al. Theorem 1 : For the single-channel path, the message waiting-time sequence { Wn(l),n2 1 ) is governed by the same statistics as the corresponding sequence for a M/G/l Let ZtCi)denote the number of messages at vi at time queueing system with Poisson arrivals of intensity h and t, where a message is counted as long as any of its packets iid service times {X,,n 2 1), distributed accordingt o is in the system (waiting or being transmitted). Then, by P { X , = mai) = g ( m ) , m = 1 , 2 , 3 * . . (12) Little's theorem we have for the limiting average message queue size W , Hence, the limiting waiting-time distribution, when PI X1E( M,) a1 = hiil < 1, where d l Mu,, is given by

+

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189

RUBIN: MESSAGE PATH DELAYS

Consequently, the average message storage capacity r e quired at zll,&f(l) equals

p p lim E ( T t ( l ) )=

bits

Theorem 2: In an N-channel path, if i is not a ladder index for {ai,l _< i _< N], then

W,Ci,

(20)

=

0,

t- m

aa.

where y(') is given by (18) , and /3 p We note that (14) reduces to the average waiting time considered in [1] when a fixed one packet message is considered (and CM2 = 0). The related expression for an exponentially distributed message length results when C M= ~ 1. The average waiting time increases linearly with C"2. When considering exponentially distributed message lengths as in (5a) ,g ( m )is given by (5b), and subsequently C d is obtained to be

C"2

=q =

exp ( - p a ) .

(21)

Here, also dl =

E(M,) al = p-lal = [l - exp ( --pa)]-lul.

(22)

for each n 2 1; i 2 2. In particular, we note (as in [l, corollary 13) that for any ladder channel ki, 2 5 k ; 5 m, and any n 2 1, we = O } . However, in the have { WnJki)= O } only if ( WnJ1) present case Wn,$l) > 0 for any j > 1, since every packet (except the first one) has to wait for the packet leader to be served first. Hence we have deduced the following conclusion. Corollary 2: For any ladder channel k;, 2 k i 5 m, any n 2 I, we have { @n(ki) = 0 ) only if { Fn(l) = 0).Also, W n , i ( k i ) > 0 for any j > 1, so that any nonleading packet has a positive waiting time for any ladder channel. The overall nth message waiting time over the first k channels is defined as