Packet Switching in a Multiaccess Broadcast Channel: Performance Evaluation

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processing, intersymbol interference, data compression, and cryptog- Dr. Hellman is a member of Tau Beta Pi and Eta KappaNU. He raphy. He has consulted in the areas of communications, informa is President of theSan Francisco Section’s IEEE Information tion theory, and general engineering forvarious industries and Theory Group’s Chapter, and was Publications Chairman for the laboratories. He is the liaison at Stanford for its Industrial Aaliates 1972 Information Theory Symposium. He is an Associate Editor Program in Information Systems, a program designed to increase for Communication Theory of the IEEE TRANSACTIONSON COM- interaction between industry and the university. MUNICATIONS.

Packet Switching in a Multiaccess Broadcast Channel: Performance Evaluation

Abstract-In this paper, the rationale and some advantages for and methods for the evaluation and optimization of the multiaccessbroadcast packet communication usingsatellite and channelperformance of aslotted ALOHA system. The ground radio channels arediscussed. A mathematical model is formulated for a “slotted ALOHA” random access system. Using this problem of performanceevaluation is addressedin this model, a theory is put forth which gives a coherent qualitative inter- paper. In [l], we present dynamic channel control pro- pretation of the system stability behavior which leads to the defini- cedures as solutions to some of the issues considered herein. tion of a stability measure. Quantitative estimates for the relative In this paper, the rationale for multiaccess broadcast instability of unstable channels are obtained. Numerical results are packet communication is first discussed. The mathematical shown illustratingthe trading relationsamong channel stability, throughput, and delay. These results provide tools for the perform- model to be considered is then described. Following that, a ance evaluation and design of an uncontrolled slotted ALOHA sys- theory is proposed which explains the dynamic and tem. Adaptive channel control schemes are studied in a companion stochastic channel behavior. In particular, we display the paper. delay-throughput performance curves obtained under the assumption of equilibrium conditions [6]. We then demon- INTRODUCTION strate that a slotted ALOHA channel often exhibits “un- N THIS and a forthcoming paper [l], a packet switch- stable behavior.” A stability definition is proposed which I ing technique based upon the randomaccess concept of characterizesstable and unstable channels. Astability the ALOHA System [Z] will be studied in detail. This measure (FET) is then defined which quantifies the technique, referred to as slotted ALOHA random access, relative instability of unstable channels. An algorithm is enables efficient sharing of a data communicatiori channel given for the calculation of FET. Finally, numerical results by a large population of users, each with a bursty data are shown which illustrate the trading relationsamong stream. This packet switching technique may be, applied channel stability, channel throughput, and average packet tothe use of satelliteand groundradio channels for delay. Our main concern in this paper is the consideration computer-computer and terminal-computer communica- of the stabilityissue and its effecton the channel through- tions, respectively’ [3]-[10]. The multiaccess broadcast put-delay performance. capabilities of these channels render them attractive solutions to two problems: 1) large computer-communication MULTIACCESS BROADCAST PACKET networkswith nodes distributed over wide geographic COMMUNICATION areas,and 2) largeterminal access networkswith po- Rationale tentially mobile terminals. The objective of this studyis to develop analytic models For almost a century, circuit switching dominated the design of communication networks. Only with the higher Paper approved bythe Associate Editor for Computer Com- speed and lower cost of modern computers did packetcom- munication of the IEEE CommunicationsSociety for publication munication become competitive. It was not until approxi- after presentation at the 7th Hawaii International Conference on System Sciences, Honolulu, Hawaii, January 8-10, 1974. Manuscript mately 1970 that the computer (switching) cost dropped received June 30, 1974; revised September 30, 1974. This research below the communication (bandwidth) cost in a packet was supported by the Advanced Research Projects Agency of the Department of Defense under Contract DAHC 15-73-C-0368. switching network [ll]. This also marked the first ap- L. Kleinrockis withthe Department of Computer Science, pearance of packet switched computer-communication University of California, Los Angeles, Calif. 90024. S. S. Lam is with the IBM Thomas J. Watson Research Center, networks [Z], [12]. Yorktown Heights, N. Y. 10598. Circuit switching is relatively inefficient for computer XLEINROCK AND LAM: PACKET SWITCHING 41 1

communications, especially over long distances. Measure- stations covered by the transponder beam. Thus, a satel- mentstudies [13] conducted on time-sharing systems lite channel (consisting of both carrier frequencies) pro- indicate that bothcomputer and terminal datastreams are vides a completely connected network topology for all bursty. Depending on the channel speed, the ratiobetween earth stations covered by the transponderbeam. the peak andthe average datarates may be as high: Consider the use of packet communication in a com- as 2000 to 1 [SI. Consequently, if a high-speed point-to- puter-communication networkenvironment tosupport point channel is used, the channel utilization may be ex- large populatons of (bursty) users over a wide area. We tremely low since the channel is idle most of the time. On can then identify and summarize the following advantages the other hand, if a low-speed channel is used, the trans- of satellite and ground radio channels over conventional mission delay is large. wire communications. The above dilemma is caused by channel users imposing 1) Elimi?lation of ComplexTopological Design and bursty random demandson their communication channels. RoutingProblems: Topological design and routing prob- By the law of large numbers in probability theory, the lems are verycomplex in networks with a large population total demand at any instant from a large population of of users. Existing implementations suitable for a (say) 50 independentusers is, with high probability, approxi- node network may become totally inappropriate for a 500 mately equal to the sum of their average demands (i.e., a node network required to perform thesame functions nearlydeterministic quantity).Thus, if a channel is [21]. On theother hand, ground radio andsatellite dynamically shared in some fashion among many users, channels used in themultiaccess broadcast mode provide a the required channel bandwidth to satisfy a given delay completely connected network topology, since every user constraint may be much less than if the users are given may access any other user covered by the broadcast. dedicated channels. This concept is known as statistical load 2) Wide Geographical Areas: Wire communications be- averaging and has been applied in many computer-com- come expensive over long distances (e.g., transcontinental, munication schemes to various degrees of success. These transoceanic). Even on a local level, the communication schemes include: polling systems [14], loop systems [15], cost, for an interactive user on an alphanumeric console asynchronous time division multiplexing (ATDM) [lS], over distances of over 100 miles may easily exceed the cost andthe store-and-forward packet switching concepts of computation [2]. On the other hand, satellite andradio [17]-[19] implemented in the ARPA network [lz]. communications are relatively distance independent, and We are currently facing an enormous growth in com- are especially suitable for geographically scattered users. puter networks [20]. To design cost-effective computer- 3) Mobility of Users: Since radio is a multiaccess broad- communication networks for the future, new techniques cast medium, it is possible for users to move around freely. are needed which are capable of providing efficient high- This consideration willsoon become importantin the speed computer-computer and terminal-computer com- development of personal terminals in future telecommuni- munications in a large network environment. The applica- cation systems [22] as well as in aeronautical and mari- tion of packet switching techniques to radio communica- time applications [23]. ti.on (both satellite andground radio channels) appears to 4) LargePopulation of Activeand Inactive Users: In provide a solution. wire communications, the system overhead usually in- Radio is a multiaccess broadcast medium. That is, a creases with the number of users (e.g., polling schemes). signal generated by a radio transmitter may be received The maximum number of users is often bounded by some over a wide area byany number of receivers. This is hardware limitation (e.g., the fan-in of a communications referred to as the broadcast capability. Furthermore, any processor). In radio communication, since each user is number of users maytransmit signals over the same merely represented by an ID number, the number of channel. This is referred to as the multiaccess capability. active users is bounded only by the channel capacity and (However, if two signals at the same carrier frequency thereis no limitation tothe number of inactive (but overlap intime at a radio receiver', we assume that potentially active) users beyond that of a finite address neither is received correctly. This destructive interference space. is the key issue in studying the multiaccess radio channel 5) Flexibility in SystemDesign: A radiopacket com- used in a packet switching mode.) Thus, a single ground munication system can become operational with two or radio channel provides a completely connected network three users. The size of the user population can be in- topology for a large number of nodes within range of each creased up to the channel capacity. More users can be other. Similarly, a satellite transponder in a geostationary accommodated by increasing the radio channel band- orbit above the earth acts asa radio repeater. Any number width. In other words, the communication system can be of earth stations may transmitsignals up to the satellite atexpanded or contractedwithout major changes inthe one carrier frequency (the multiaccess channel). Any basic system design and operational schemes. signal received by the satellite transponderis beamed back 6) StatisticalLoad Averaging: Wire communication to earth at another frequency (the broadcast channel). links are more efficiently utilized in a store-and-forward This broadcasted signal may be received by allearth packet switched network than in a circuit switched network. However, atany instant, there may be unused 1 This event will be referred to as a channel collision. channel capacity in some parts while congestion exists in 412 APRIL COMMUNICATIONS, ON IEEE TRANSACTIONS 1975

otherparts of the network. The application of packet “slott,ed ALOHA.” In this scheme, the users transmit switching techniques to a single high-speed satellite or newly generatedpackets into channel time slots inde- radio channel permits the total demand of all user input pendently. In the event of a channel collision, the collided sources to be statistically averaged at the channel. Note packets areretransmitted after random retransmission also that each usertransmits dataat the wide-band delays. (See Fig. 1.) The channel capacity of aslotted

channel rate. ALOHA channel was shown to be l/e N 36 percent [25], ’ 7) Multiaccess Broadcast Capability: This capability in To achieve a channel throughput rate larger than the radio communication may be useful for certain multipoint- 36 percent limitation, various other multiaccess broadcast to-multipoint communication applications. packet swiching schemes havebeen proposed totake advantage of special systemand traffic characteristics. The Multiaccess Channel Model The reader is referred to the references [3], [7], [26] for Consider aradio communication systemsuch as a description of t,hese schemes. packet switched satellite system [5>[10] or the ALOHA Consider slotteda ALOHA channel. The channel System [2]. In each case, there is a broadcast channel for input in a t,ime slot, is defined to be a random variable point-to-multipoint communication and a multiaccess representing t,he tot,al number of new packets transmitted channelshared by alarge number of users. Since the by all users in that time slot. Assuming stationary condi- broadcastchannel is used by a single transmitter, no tions, the channel input rate X is the average number of transmission conflict will arise. All nodes covered by the new packet transmissions per time slot.. The channel trafic radio broadcast canreceive on the same frequency,picking in a time slotis defined to be a random variable represent- out packetsaddressed to themselves and discarding ing the tot(a1number of packet transmissions (both new and packets addressed to others. previously collided packets) by all users in that bime slot. The problem we are faced with is how to effect time- Assuming stationary conditions, the channel traffic rate sharing of the multiaccess channel among all users in a G is the average number of packet transmissions per time fashion which produces an acceptable level of perform- slot. The channel’ throughput (or output) in a time slot is ance. As soon as we introduce the notion of sharing in a defined to be a random variable representing the number packet switching mode, we must be prepared to resolve (0 or 1) of successful packet transmissions in t,hat time conflicts which arise when simultaneousdemands are slot. Assuming stationary conditions, the channel through- placed upon the channel. There are two obvious solutions put (output) rat’e Soutis t,he probability of exact,ly one td this problem: the first is to form a queue of conflicting packet transmission in a channel time slot. demands and serve them in some order; the second is to The retransmissiondelay (RD) incurredby an un- (‘lose” any demands which are made while the channel is successful packettransmission may be regarded as the in use. The former approach is taken in ATDM and in sum of a deterministic component (R)and a random com- store-and-forward networks assuming that storage may be ponent.The random component is necessary since if provided economically at the point of conflict. The latter collided packets are retransmitted after the same deter- approach is adopted in the ALOHA System random access ministic delay, they mill collide again for surk. In a ground scheme; in this system, in fact, all simultaneous demands radio system, RD corresponds to thepositive acknowledg- made on the radio channel are lost. ment time-out interval [a]. In a satellite system, since Letus define channelthroughput rate Xout to be the each channel user listens to the satellit,e broadcast., one average number of correctly received packet transmissions round-trip propagation time after transmittsing a packet per packet transmission time (assuming stationary con- he knows .whether he was successful or if a channel colli- ditions). We also define channelcapacity X,,, to be the sion occurred. In this case, the deterministic component maximum possible channel throughput rate. The channel corresponds to around-trip satellite propagation delay. capacity of a pure ALOHA multiaccess channel was shown We shall assume a noise-free channel such t,hat a packet is by Abramson to be 1/2e s 18 percent for a fixed packet received incorrectly if and only if it suffered a channel size [Z]. Under similar assumptions, Gaarder showed that collision. In [SI, a uniform probabilitydist,ribution is a pure ALOHA channel with a fixed packet size is always assumed for the random component of RD such that superior (interms of channelcapacity) to one with each userretransmits a previously collided packet at different packet sizes [24]. random during one of the next K slot,s (each such slot Roberts suggested that the channel may be slotted by being chosen with probability l/K). Thus, retransmission requiringall users to synchronize2 the leading edges of will take place either R + 1, R + 2, - - ‘or R + K slots their packet transmissions to coincide with an imaginary afterthe previous t,ransmission. This is said to be the timeslot boundary at the multiaccessed radio receiver uniform retransmission randomization scheme. Under this [25]. The duration of a channel time slot is chosen to be scheme, equilibrium throughput-delay tradeoffs have equal to a packet transmission time. The resulting scheme been obtained for aslotted ALOHA channel witha will be referred to as “slotted ALOHA random access” or Poisson input source (the infinite populationmodel). Such throughput-delay contours are shown here in Fig. 2 * The problem of synchronizing channel users is a nontrivial one. for differentvalues of K. Note that theminimum envelope It will not be addressed in this paper. of these contours defines the optimum channel perform- KLEINROCK AND LAM: PACKET SWITCHING 413

given by S, K, and DA as thechannel operating point, since USER 1 this is the desired channel performance given S and K.)

USER 2 This observation suggests that the assumption of equi- , librium conditions adoptedin most previous analytic USER 3 ...... models [4]-[7] may not be valid. * __------.x In order to study the dynamic behavior of these chan- USER 4 TIME nels, simulations were performed for the infinite popula- - tion model [lo]. Eachsimulation run was observed to SUCCESSFUL PACKETTRANSMISSION behave in the following manner. Starting froman initially TRANSMISSION CONFLICT emptysystem, the channel stays in equilibrium at the ,--\RANDOM RETRANSMISSION DELAY channel operating point for a finite period of time until Fig. 1. Slotted ALOHA random access. stochasticfluctuations give rise to some high channel traffic rate which reduces the channel throughputrate which in turn further increases the channel traffic rate. As this vicious cycle *continues, the channel becomes inun- datedwith collisions andretransmissions. At the same time, the channel throughput ratevanishes rapidly to zero. This phenomenon will be referred to as channel saturation. Thus, we realize that the equilibrium throughput-delay tradeoffs are notsufficient to characterize the performance of the infinite population model. A more accurate measure of channel performance must reflect the trading relations among channel stability, throughput and delay. A mathematical model with a simpler structure than that used in n L 100 - [SI will be defined below. This model is similar to the one studied byMetcalfe [4]. Using this model, the concepts of channel saturationand stability in aslotted ALOHA random access channel have been characterized [SI, [lo]. - 40 STABILITY-THROUGHPUT-DELAY 30 - TRADEOFF PERFORMANCE In this section, a Markovian model is first formulated for a population of M channel users. The variable M is assumed to be large and may be either finite or infinite. A theory is then proposed which characterizes the instability 10 I I I I I I I, I 0 .05 .10 .15 20 25 .30 .35 lle .4 phenomenon in the following ways. S THROUGHPUT IPACKETSISLOTJ 1) Stable and unstable channels are defined. Fig. 2. Equilibriumthroughput-delay tradeoff. 2) In astable channel, equilibrium throughput-delay results (as shown in Fig. 2) are achievable over an infinite ance. These results correspond to the use of a 50 KPBS time horizon. In an unstable channel, such channel per- satellite channel, 1125 bits per packet,and a satellite formance is achievable only for some finite time period round-trip propagation delay of 0.27 s for all users. Thus before the channel goes into saturation. R is equal to 12 slots and there are44.4 slots inone second. 3) For unstable channels, a stability measure is defined (These numbers will be assumed throughout this paper.) and anefficient computational procedure for its calculation In Fig. 2, D represents the average packet delay in slots. is given. Note that thechannel input rate S is equal to thechannel 4) Using the abovestability measure, the stability- throughputrate Soutunder the assumption of channel throughput-delay tradeoff for unstable channels is ex- equilibrium. The channel capacity S,,, approaches l/e in amined. the limit as K --$ w . For K = 15, it is almost there. For values of K between 8 and 15, the equilibrium throughput- The Markovian Model delay tradeoffs are very close to the optimum perform- We consider aslotted ALOHA channel with a user ance envelope over a wide range of S. population consisting of M users. Each suchuser can be in The analyticresults presented so far arebased upon the one of two states: blocked or thinking. Inthe thinking assumption that the channel is in equilibrium. Referring state, a user generates and transmits a new packet in a to Fig. 2, we see that given S and K (say K = 40) , there time slot with probabilityu. A packet which had a channel are two possible equilibrium solutions for D! They cor- collision and is waiting for retransmission is said to be respond to asmall delay value DA and a muchlarger backlogged. The retransmissiondelay RD of each back- delay value Dg. (We shall refer to the equilibrium point logged packet is assumed to be geometrically distributed, 414 IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 1975

i.e., each backlogged packetretransmits in the current sitates a statedescription consisting of the channel history time slot with probability p. Assuming bursty users, we for at least R consecutive time slots. The difficulty in must have p >> u. From the timea user generates a packet mathematical analysis using such a state description was untilthat packet is successfully received, the user is illustratedin [lo]. However,simulation results have blocked in the sense that he cannot generate (or accept shown that the slotted ALOHA channel performance (in from his input source) a new packetfor transmission. terms of averagethroughput and delay) is dependent Let Ntbe a random variable (called the channel backlog) primarily upon the average retransmission delay (m)and representing the total number of backlogged packets at quite insensitive tothe exact probabilitydistributions time t. The channel input rate at time t is St = (M - considered [lo]. In order to use the'analytic resultsof the Nt)u. Note that St decreases linearly as Nt increases. The Markovian model here to predict the throughput-delay vector (Nt,St)will be denoted as thechannel state vector. In performance of a slotted ALOHA channel with nonzeroR, this context, both M and u may be functions of time. We it is necessary to use a value of p in the Markovian model shall assume M and u to be time-invariant unless stated which gives the same m. For example, to approximate a otherwise. In this case, ' Nt is a Markov process (chain) slotted ALOHA channel with uniform retransmission ran- with stationary transition probabilities and serves as the domization, we must let state description for the system. The discrete state space will now consist of the set of integers { 0,1,2, - - .,M}. The one-stepstate transitionprobabilities of Nt are,for i = 0,1,2,. * .,M,

j5i-2 I O' ip(1 - p)"-'(l - u)M--i

(1 - p)i(M - i)a(l - ,)M--i--l+ [l -.ip(l - p)i--'](l - u)M--i j = i pii = Prob [Nt+-l = j I Nt = i] = (M - i)u(l - u)M--(--'[l - (1 - p)i] j=i+l

The assumption that RD has a memoryless geometric We define the length of time for which a packet is back- distributionpermits a simple state description for the logged to be the backlog time of the packet and denote the mathematicalmodel. However, this assumption implies average backlog time by Db. To obtain the average packet that RD has a zero deterministic component (R = 0).In delay (as defined in [SI), we must add to Db, R + 1 time a satellite channel this obviously representsan approxima- slots, which represent the delayincurred by each successful tion. (However, it may be physically realizable in radio transmission. Thus, we have communicationsover short distances in which channel D=Db+R+l. (4) propagation delays are negligible compared to a packet transmissiontime.) A (geostationary)satellite channel Numericalresults in thispaper will beexpressed in has a round-trip propagation delay of 0.27.~~which neces- terms of K (rather than p) through use of (3) and (4) for KLEINROCK AND LAM: PACKET SWITCHING 415 comparison with previous results for channel performance [SI. The Theory

Conditioning on Nt = n, the expected channel through- S put Sout(n,u)is theprobability of exactly one packet 0 I transmission in the tth time slot. Thus, Fig. 3. Throughput surface above the (n,S)plane.

Sout(n,u)= (1 - p)"(M - n)u(l - ~)~--n--l n + np(1 - p)"-'(l - u)~-". (5) For the infinite population model, i.e., in the limit asM ~0 and u 0 such that Mu = S is finite and the channel input is Poisson distributed at the constant rate S, the above equation reducesto

Sout(n,S)= (1 - p)"S exp (-SI + np(1 - p)"--lexp (-AS). (6) This expression is very accurate evenfor finite M if u << 1 and if we replace S = Mu by S = (M - n)u. We assume that the condition u << 1 (which implies bursty users) is always satisfied in problems of interest to us. In Fig. 3, for a fixed K we sketch Sout(n,S)as a three- dimensional surface abovethe (n,S)plane. Note that there is an equilibrium contour in the (n,S)plane defined as the locus of points on which the channel input rate S is equal to the expectedchannel throughput Sout(n,S)given by " (6). In thecrosshatched region enclosed by theequilibrium 0 .IO .20 .30 contour, Sout(n,S)exceeds S;elsewhere, S is greater than CHANNEL INPUT (PACKETWSLOT) Sout(n,S).In Fig. 4, a family of equilibrium contours for Fig. 4. Equilibrium contours in the (n,S)plane. various K are displayed. We see that if we increase the average retransmission delay (by increasing K or equiva- lently decreasing p), the equilibrium contour moves up- wards. Weshow below that these equilibrium contours play a crucial role in determining the stabilitybehavior of the channel. Given an equilibrium contour in the (n,S) plane, we first consider the dynamic behavior of the channel subject to time-varying inputs using a fluid approximation interpretation. The following example serves to illustrate the underlying concepts. n Consider the case in which u is constant while M = M(t) is a function of time as shown in Fig. 5. We use the fluid approximation for the trajectoryof the channel state vector (Nt,St)in the (n,S) plane as sketched in Fig. 6. Recall that St = (M - Nt)u. The arrowsindicate the "fluid" flow direction which depends on the relativemagni- tudes of theinstantaneous channel throughput rate Sout(n,S)and the channel inputrate S. Two possible cases are shown corresponding to different values of the amplitude M3, of the input pulse in Fig. 5. The solid line S (Case 1) representsa trajectory which returnsto the Fig. 6. Fluid approximationtrajectories. original state on the equilibrium contour despite the input pulse. The dashed line (Case 2) represents a less fortunate eventually, the channel '(fails" as a result of an increasing situation inwhich the decrease in thechannel input rateat backlog and a vanishing channel throughput. time tz is not sufficient to bring the trajectory back into theThe above exampledemonstrates channel saturation 416 IEEE TRANSACTIONSAPRIL ON COMMUNICATIONS, 19'75

n n ditions under which the slotted ALOHA channel with a CHANNEL I SATURATION stationary input (constantM and u) can go into saturation CHANNEL POINT as a result of statistical fluctuations. OPERATING CHANNEL Assume that M and u are constant. The trajectory of (Nt,St) is constrained to lie on thestraight line X = (A4 - n)u called the channel load line which intercepts the "c

n-axis at n = M and has a slope equal to - I/U. We now "0 propose the following definition for characterizing stable s "0 and unstable channels. SO SO The Stability Definition: A slotted ALOHA channel is (a1 A STABLE CHANNEL (b)AN UNSTABLE CHANNEL said to be stable if its load line intersects (nontangentially) n n the equilibrium contour in exactly one place. Otherwise, the channel is said to be unstable. Examples of stable and unstable channels are shown in Fig. 7. Arrows on the channel load lines indicate directions of fluid flow given by the fluid approximation. In other words, the arrows pointin the direction of increasing backlog size if X > Sout(n,X) and in the direction of decreasing backlog size if Xout(n,S)> X. -S Each channel load line may have one or more equilib- (c) ANUNSTABLE CHANNEL (d) ANOVERLOADED CHANNEL rium points. A point on the load line is said to be a stable Fig. 7. Stable and unstablechannels. equilibrium pointif it acts as a "sink" with respect to fluid flow. It is a globally stable equilibrium pointif it is the only expected channel backlog can then be obtained from stable equilibrium point on the channel load line. Other- fl wise, it is a locally stable equilibrium point. (Each stable M Sout= C Sout(n,u)P, (7) equilibrium point is identified by a dot on channel load n=O lines in Fig. 7 except in Fig. 7(c), where one of the stable and equilibrium points is at n = 00 .) An equilibrium point is said to be an unstable equilibrium point if fluid flow ema- M 8 = nP,. (8) nates from it. Thus, the channel state Ntsitting on such a n=l point will drift away from it given the slightest perturba- Numerical results haveshown that these valuesof Soutand tion. The stability definition given above is equivalent to for a stable channel are closely approximated by the defining a stable channelto be one whose channel load line m equilibrium S, and no at the channel operating point, and has a globally stable equilibrium point. also by the equilibrium throughput-delay values in Fig. 2 In Fig. 7 (a), we show the channel load line of a stable for the infinite population model. For example, suppose channel. The globally stable equilibrium point on the load K = 60, M = 200, and 1/u = 536.1 ; the equilibrium line, (no,So),will be referred to as the channeloperating channel throughput rate at the cha,nnel operating point is point. If M is finite, a stablechannel can alwaysbe So = 0.346. In Fig. 9 below (to be described later), we achieved by using a sufficiently large K (see Fig. 4). Of see that the steady-state channel throughput rate com- course, a large K implies that theequilibrium backlog size puted by using (7) is Sou,= 0.344. For the same example, nois large; the corresponding average.packet delay maybe is calculated to be 15.4 slots. By Little's result [27], the too large to be acceptable. Since the Markov chain Nt has fl average backlog time is a finite state space and is irreducible (assuming p,u > 0) , ,. a stationary probability distribution always exists [27], --=--fl 15.4 D b- - 44.8 slots. [28]. The stationary probability distribution { P,) ,,OM of Sout 0.344 Nt can be computed by solving the following set of linear simultaneous equations Applying (4)) we get D = 44.8 + 13 = 57.8 slots. Now given X, = 0.346, the K = 60 equilibriumthroughput- M delay contour for the infinite population model [SI gives Pj = pipij j = 0,1,. *,&I D = 56.5 slots. i=O In Fig. 7(b), we show the channel load line of an un- and stablechannel. The point (no,#,) is again the desired M channel operating point since it yields the larger channel cpi=1 throughput and smaller average packet delaybetween the i-0 two locally stable equilibrium points on the load line. In where the state transition probabilities pij are given by fact, the other locally stable equilibrium point, having a (1). The steady-state channel throughput rate Soutand huge backlog and virtually zero throughput, corresponds KLEINROCK AND LAM: PACKET SWITCHING 417 to thechannel saturation state; itwill be referred to as the 0.5 channelsaturation point. Although it has a stationary 0.4 probabilitydistribution, Nt will ‘(flip-flop” between the two locally stableequilibrium points in the following 0.3 manner. Starting from an empty channel (NO = 0) quasi- stationary conditions will prevail at the operating point (no,S,).The channel, however, cannot maintain equilib- 0.2 I rium at this point indefinitely since Ntis a random process; sI- 3 that is,with probability one, the channel backlog Nt VI crosses the unstable equilibrium point n, in a finite time, tY 0

and as soon as it does, the channel input rate S exceeds I2 0.1 Sout(n,X).Under this condition, Nt will drift toward the w d saturation point. Although there is a nonzero probability I- that Ntmay return below n,, all our simulations show that 2 0 the channel state Nt accelerates up the channel load line 3 8 0.05 producing an increasing backlog and a vanishing through- I J put rate. Since thesaturation point is a locally stable W z z equilibrium point, quasi-stationary conditions will prevail a I there for some finite (but probably very long) timeperiod. 0 In this state, the communication channel can be regarded as having failed. (In a practical system, external control should be applied at this point to restore proper channel operation.) Thus, thetwo locally stable equilibrium points on the load line of an unstable channel correspond to the

channel being “up” or “down”. An unstable channel may 0.01 I I I I I be acceptable if the average channel up time is large and 50 60 70 10080 90 M external control is available to bring the channel back up NUMBER OF USERS whenever it goes down. Fig. 8. Channel performance versus M at K = 10 and So = 0.36. In Figs. 8 and 9, we see how, as the number of channel users M increases, an originally stable channel becomes 0.5 unstable although the channel input rate X, at theoperat- ing point remains constant (by reducing CT) . (These results 0.4 are obtained by first solving for the stationary probability 0.3 distribution of NCand then applying (7) and (8).) For So = 0.36 and K = 10, we see that as M exceeds 80, the stationary channel throughputrate decreases andthe 0.2

I average packet delayincreases very rapidly withM. Using sI- the K = 10 equilibrium contour in Fig. 4, the maximum ev)

value of M that is possible without making the channel klY 0 load line intersect theequilibrium contour more than once 3 0.1 W is determined(graphically) to be Mmax= 79, which I- K exactly gives the knees of the curves in Fig. 8. This ex- I- 3 & cellent agreement provides the motivation for the stability I 0 definition proposed above. In Fig. 9, by using a larger value 3 8 0.05 of K ( =60), a larger MmBxis possible. Note, however, I I- -I that the average packet delay (h.56 slots) for K = 60 is Y z z much larger than theaverage packet delay(=36 slots) for Q K = 10. 0 30 Given K and So,M,,, can be obtained graphicallyfrom the equilibrium contours such as shown in Fig. 4. In Fig. 10 we show M,,, as a function of K with So fixed at the maximum possible value given K. Note the linear relation- ship between M,,, and K for the values shown. In Fig. 11, we illustrate how an originally unstable channel can be 0.01 1 I I I I rendered stable by using a sufficiently large K. 2oI170 180 190 200 210 220 In Fig. 7(c), we show the channel load line of an in- NUMBERM OF USERS finite population model. This is an unstable channel since Fig. 9. Channel performance versus M at K = 60 and So = 0.346. 418 IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 1'375

stable equilibrium point in thiscase is the channel satura- 300 r tion point! Thus, this represents an "overloaded" channel as a result of bad system design. To correct this situation, the number of active users M supported by the channel should be reduced. From now on, a stable channel will always refer to the load lie depicted in Fig. 7(a) instead of Fig. 7 (d) . Let us summarize the major conclusions in the above discussion. 1) The steady-state throughput-delay performanceof a 0 I I I I I 1 I I I I stablechannel isclosely approximatedby its globally 0 10 20 30 40 50 60 70 80 90 100 K stable equilibrium point and by the equilibrium through- Fig. 10. M,,, versus K. put-delay results for the infinite population model. 2) In an unstable channel, the throughput-delay per-

0.5 . 500 formance at a locally stableequilibrium point can be achieved only for some finite time period. 0.4 ' 400 A Stability Measure 0.3 ' 300 From the above discussion and referring to Fig. 7(b) 9 the load line of an unstable channel can be partitioned 0.2 200 into two regions. The safe region consisting of the channel

I sc states (0,1,2,...,nc)and the unsafe region consisting of 3 thechannel states {n, 1,- -,M). A good stability v)+ - + - W E Y s measure (for these unstablechannels!) is the average time 0 2 g 0.1 >. 100 to exit into the unsafe region starting from a safe channel Y 90 I- w9 state. To be exact, we define FET to be the average jirst 2 n 80 exit time into the unsafe region starting from an initially + & 70 z = I 0 empty channel (NO 0). Thus, FET gives an approxi- 0 3 260 matemeasure of theaverage up time of an unstable 0.05 2 50 I 2 channel. Below we derive the probability distributions and + w 2 W >40 expected values of such first exit times. The derivations z z a are based upon well-known results of first entrance times 5 30 in the theory of Markov chains with stationary transition probabilities [ZS], [30].

20 Consider the Markovian model with constant M and CT,where M may be infinite. Ntis a Markovprocess (chain) with stationary transition probabilities { pij) given by (1) or (2). Define the randomvariable Tij to be the numberof transitions which Nt goes through until it enters statej for 0.01 10 I I I I I 70 80 11090 100 120 the first time starting from state i. The probability distri- K bution of Tii (called the Jirstentrance probabilities from Fig. 11. Channel performance versus K at M = 250 and l/~= 675. state i to state j) may be defined as

m=O

m=l fij(m) = Prob [Tij = m] = I"pij

~~ ~~ n = * is a stable equilibrium point. In fact, since Nt has The state space s for Nt consists of the setof nonnegative an infinite statespace and S > Sout(n,S)for n > n,, a integers {0,1,2,. + -,n,, n, + 1,. - .,M) which is parti- stationary probability distribution does not exist for Nt. tioned into the safe region (0,1,2, - - -,nc) and the unsafe (See, for example, [29, pp. 543-5461 for such a proof in a region { n, + 1,. * ,M). Now consider the modified state queueingcontext.) space S' = { 0,1,2, - ,nc,n,) where n,absorbing is an state The channel loadline shown in Fig. 7(d) is stable such that Nt is now characterized by the transition prob- according to the stability definition. However, the globally abilities KLEINROCK AND LAM: PACKET SWITCHING 419

no Pi = 1 + p

no 2-nc+l T,2 = 2?li - 1 + PijF i = 0, 1,. -*,n,. (13) j=O i=n.j=O21, , .,n, Equation (12) forms a set of n, + 1 linear simultaneous equationsfrom which ( !f?i]i=On=can be solved andthe .. stability measure FET ( = determined. After { i-onc a =3 = nu. Pi} I havebeen found, (13) can then be solved ina similar Define therandom variable Ti to be the number of manner for { TT}i=Onc. transitions which N1 goes through before it entersthe unsafe region for the first time starting from statei in the Numerical Results safe region. Ti is called the first exit time from state i. The With the stability measure defined above, we are now probability distribution of Ti is defined to be { fi(m)}m-lm in a position to examine quantitatively the tradeoffamong which are called the first exit probabilities, It is trivial to channel stability,throughput and delay for unstable show that starting from state i (0 i i 5 n,), the first channels. Below we first give a computational procedure entrance probabilities into the absorbing state nu in the to solve for !f?i and hence, FET. We then compute these modified state space S’ arethe same asthe first exit quantities for various values of K, So,and M (correspond- probabilities into the unsafe region of S. Using (9), such ing to different channel load lines). The trading relations probabilities are given by the following recursive equation among channel stability, throughput, and delay are then C301, illustrated.

no The solution of the set of simultaneousequations in fin,(m) = pin,’S(m - 1) + Pii’fjn,(m - 1) either (12) or (13) requires invertingthe (n, + 1) by i=O (n, + 1) matrix of pij for i, j = 0, 1, - - - ,n,. When n, is m 2 1;i # nu large, this becomes a nontrivial task because of the large where number of computationalsteps and large computer 1 m=l storage requirement for the [p~j]matrix. The fact that 8(m) = pij = 0 for j i - 2 in (1) and (2) enables us to use an 0 otherwise. algorithm given in the Appendix which is very efficient in , I terms of both computer time and spacerequirements. For The above equation can be rewritten in terms of the first our purposes, this algorithm is superior to conventional exit probabilities as methods such as Gauss elimination [31] for solving linear simultaneous equations. In this algorithm, each pii is used M no exactly once and can be computed using (1) or (2) only fi(m) = pi$(m - 1) + piifj(m - 1) j=nc+l j-0 when it is needed inthe algorithm. This eliminates the need m2l;Oli

500 r I Io7 400 I DAY

300 ;i 200 1 I- -3

I MIN

103

I02 20 4080 60 100 K

Fig. 12. FET values for the infinite population model. 10 I I I I I, .2 .25 .2 3 .35 I/e

SO FET (SLOTS) Fig. 15. Stability-throughput-delay tradeoff.

M is 150. Recall that if M is finite, the channel will become stable when K is sufficiently large. As an example, we see that in Fig. 14 for M = 150, if the channel throughput rate X, is kept at approximately- 0.28 and K = 10 is used, the channel is estimated to fail once every two days onthe average. If this is an acceptable level of channel reliability, then no other channel control procedure is necessaryexcept torestart the channel whenever it goes intosaturation. However, if absolute channel reliability is required at the same ,throughput- delay performance, then dynamic channel control strategies should be adopted. Channelcontrol schemes have been studied [lo] and the results will be published in a forthcoming paper [l]. In Fig. 15, we show the optimum performance envelope 200 300 400 NUMBER OF TERMINALS M inFig. 2 as a lower bound for thethroughput-delay Fig. 13. FET versus M. tradeoff of the infinite populationmodel. This corresponds to the performance of the channel at the channel operatingpoint. However, from Fig. 7, we see that the FET (SLOTS) ' channeloperating point (n,,X,) provides noinformation regarding the stability behavior of the channel. The equilib- riumperformance given by (n,,S,) is achievable in the long run if M is small enough such that the channel is stable; elsewhere it is achievable only for some random time period whose average is estimated by our stability measure FET . In addition to the infinite population model optimum envelope, we also show in Fig. 15 two sets of equilibrium .throughput-delayperformance curves with guaranteed FET values. The first set .consists of three solid curves corresponding to an infinite population model with the stability measure FET 2 1 day, 1 hour, and 1 minute. NUMBER OF TERMINALS M = I50 Again, these results represent worst case estimates if M is actually finite. The second set consists of twodashed curves corresponding to M = 150 with FET 2 1 day and 1 hour. These results were obtained by looking up the Fig. 14. FET values for afinite user population (M = 150). values of K and X, in Fig. 12 or Fig. 14 corresponding to a KLEINROCK AND LAM: PACKET SWITCHING 42 1 fixed FET. The average packet delay was then obtained APPENDIX from Fig. 2. This figure illustrates the fundamental trade08 The algorithm below solves for the variables {ti)ioo' in among channel stability, throughput and delay. In [l], the following set of (I + 1) linear simultaneous equations, [lo], control strategies are devised to dynamically regu- late the channel usage to achieve truly stable throughput- I to = ho + pojtj (AI) delay performance close t.o the optimum performance j-0 envelope. I ti = hi + pijtj i = 1, 2,. *,I. (A21 A Design Example j-i-1 The designer of a slotted ALOHA channel is faced with the problem of deciding whether he wants 1) astable The Algorithm channel by limiting its use to a small population of users 1) Define and sacrificing channel utilization,or 2) an unst,able er = 1 channel which supports a large population of users operating at a certain level of reliability (some value of FET). fr = 0 For example, suppose K is chosen to be 10. (Note in Fig. 1 - prr 2 that K = 10 gives close to optimum equilibrium through- er-l = - put-delay performance over a wide range of channel PIJ-1 throughputrate.) Also, suppose that the cha.nne1 users have an average think time of 20 s which, for our channel numerical constants, corres,pond to 888 time slots. Now if we draw channel load lines in Fig. with a slope equal to 4 2) For i = I - 1, I - 2,- .,1 solve recursively -888, the channel is stableup to approximately 110 I channel users. For M = 110, the channel throughput rate 1 So is about 0.125 packet/slot. From Fig. 2, the average packet delay is roughly 16.5 timeslots (=0.37 s). The same channel can be used (in anunstable mode) to support 220 users at achannel throughput rate of So = 0.25 packet/slot. The averagepacket delay is 21 timeslots (=0.47 s). From Fig. 12, for K = 10 and So = 0.25, the 3) Let average up time (FET) of the channel is approximately two days for the infinite population model. Note that this value represents a lower bound for the FET of M = 220. Thus, we see that if a channel failure rate of once every two days on the average is an acceptable level of reliability, the second channel design is much more attractive than the first since the number of channel users is more than doubled at a modest increase in delay.

CONCLUSIONS Derivation of the Algorithm In this paper, the rationale and some advantages for Define broadcast packet communication have been discussed. A mathematical model was then formulated for aslotted ALOHA random access system. IJsing this model, a and theory was put forth which gives a coherent qualitative interpretation of the system stability behavior. Quanti- er = 1 tative estimates for the relativeinstability of unstable fr = 0. channels were obtained through definition of the stability measure FET. Numerical results were shown illustrating The last equation in (A2) is the trading relations among channel stability, throughput and average packet delay. These results establish tools for the performance evaluation and design of an uncontrolled slotted ALOHA system. Further'improvement in the system performance may be accomplished through adaptive control techniques studied in [l], [lo]. .

422 IEEE TRANSACTIONS ON COMMUNICATIONS, APRIL 1975

Equating the coefficients of tr and the constant terms, we F.Heart, “A systemfor broadcast communication: reservation-ALOHA,’’ in Proc. 6th Hawaii Znt. Conf. System Sciences, have Univ. Hawaii, Honolulu, Jan. 1973. [4] R. M. Metcalfe,“Steady-state analysis of aslotted and con- 1 - PII trolled ALOHA system with blocking,” in Proc. 6thHawaii = ~ Int. Conf. System Sciences, Univ. Hawaii, Honolulu, Jan. 1973. PIJ- 1 [5] N. Abramson, “Packet switching with satellites,’’ in 1973 Nat. Comput.Conf., AFZPS Conf. Proc., vol. 42. New York: hr AFIPS Press, 1973, pp. 695-702. ’---- [6] L. Kleinrock and S. S. Lam,“Packet-switching in a slotted f 1-1 - pz,z- 1 satellitechannel,” in 1973Nat. Comput. Conf., AFZPS Conf. Proc., vol. 42. New York: AFIPS Press, 1973, pp. 7OS710. [7] L. G. Roberts, “Dynamic allocation of satellite capacity through Equation (A2) can be rewritten as follows, packet reservation,” in 1973 Nat. Comput. Conf., AFIPS Conf. Proc., vol. 42. New York: AFIPS Press, 1973, pp. 711-716. 1 I [SI L. Kleinrock and S. S. Lam, “On stability of packet switching ti-1 = -[ti - hi - pijtj]. (A6) inarandom multi-access broadcastchannel,” in Proc. 7th pi,i-l j= i Hawaii Znt.Conf. System Sciences (Special Subconf. Computer Nets), Univ.Hawaii, Honolulu, Jan. 8:lO 1974. In each of the above equations, use (A3) to substitute for [9] S. Butterfield, R. Rettberg,and D. Waiden, “Thesatellite IMP for the ARPA network,” in Proc. 7th Hawaii Int. Conf. ti. We then have System Sciences (Special Subconf. Computer Nets),Univ. Hawaii, Honolulu, Jan. &IO, 1974. [IO] S. S. Lam, “Packet switchingin a multi-access broadcast channel with application to satellite communication in a computer network,” Ph.D. dissertation, Dep. Comput. Sci., Univ. Calif., Los Angeles, Mar. 1974; also in Univ. of Calif., Los I I Angeles, Tech. Rep. UCLA-ENG-7429, Apr. 1974. [ll]L. G. Roberts, “Data by the packet,” ZEEE Spectrum, vol. 11, L (C piiej)tr - C pijfi]. pp. 46-51, Feb. 1974. j=.i j= i [I21 L. G. Roberts and B. D. Wessler, “Computer network development to achieve resource sharing,” in 1970 Spring Joint Comput. Equating the coefficients of tl and the constant terms,we Conf., AFIPS Conf.Proc., vol. 36. Montvale, N. J.: AFIPS Press, 1970, pp. 543-.549. get [I31 P. E. Jackson and C. D. Stubbs, “A study of multiaccess com- i I puter communications,” in 1969Spring Joint Comput. Conf., AFZPSConf. Proc., vol. 34. Rlontvale, N.J.: AFIPS Press, 1969, pp. 491-504. [I41 J. Martin, SystemsAnalysis jor DataTransmission. Engle- wood Cliffs, N. J.: Prentice-Hall, 1972. 1 Z [I51 J. I<..Pierce, “Network for block switching of data,” in ZEEE Conv. Rec., New York, Mar. 1971. [16] W. W. Chu, “A study of asynchronous time division multiplexing for time-sharing computer systems,” in 1969 Fall Joint From (A4), (A5), and (A7), ei and fi (i = I - 2, I - 3, Comput. Conf., AFZPSConf. Proc., vol. 35. Montvale, N. J.: AFIPS Press, 1969, pp., 669-678. - * ,l,O) can then be determined recursively. [I71 P. Baran, “On distrlbuted communications XI.Summary We next solve for tI. Equation (A3) is used tosub- overview,” Rand Corp., Santa Monica, Calif., Memo. RM- 3767-PR,, Aug. 1964. stitute for ti in (Al).,which then becomes [18] L. Kleinrock, Communication Nets: Stochastic Message FZow and Delay. New York: McGraw-Hill, 1964 (out of print); reprinted I. Z by New York: Dover, 1972. eotI + fo = ‘ho + ccpojej)tr + c pojfi. [I91 11. W. Davies, “The princip)es of a data communication net- j-0 j=O work for computers and remote peripherals,” in Proc. Int. Fed. Information Processing Congr., Edinburgh, Scotland, 1968, pp. Solving for tI in the above equation, we have Dll-Dlfi. [20] P. Wright, “Facing a booming demand for networks,” Datama- tion, vol. 19, pp. 138-139, Nov. 1973. [21] H. Frank,M. Gerla, and W. Chou,“Issues in the design of large distributed computer communication networks,” in Proc. Xat. Telecommunications Conf., Atlanta, Ga., Nov. 26-28, 1973. [22] L.G. Roberts,“Extensions of packet communication technology to a hand held personal terminal.,” in 1972 Spring Joint Con~put.Conf., AFIPS Conf.Proc., vol. 40. Montvale, N. J.: AFIPS Press, 1972, pp. 295-298. 1231 In Inst. Elec. Eng.(London) Proc. Znt. Conf. SatelliteSystems for Mobile Communications and Surveillance, Mar. 13-15, 1973. [24] N.T . Gaarder, “ARPANET satellite system,” ARPA Network Finally, ti(i = 0,1, 2,--.,I - 1) can be obtained from Inform.Center, Stanford Res. Inst., Rlenlo Park, Calif., ASS (A3),since e;, fi, and tI are all known. The derivationof the Note 3 (NIC 11285), Apr. 1972. [25] L. G. Roberts, “ALOHA packet system with and without slots algorithm is now complete. andcapture,” ARPA NetworkInform. Center, Stanford Res. Inst., Menlo Park, Calif., ASS Note 8-(NIC 11290), June 1972. [26] L. Kleinrock and F. A. Tobagi, “Carrier-sense multiple access for packet switched radio channels,”in Proc. Int. Conf. Com- REFERENCES n~unications,Minneapolis, Minn., June 1,974. [27] L. Kleinrock, QueueingSystems, Vol. Z, Theory,Vol. ZZ, Corn- [I] S. S. Lam and L. Kleinrock, “Packet switching in a multiaccess puterApplications. New York: Wiley-Interscience, 1975. broadcast channel: dynamic control procedures,” ZEEE Trans. [28] E. Parzen, Stochastic Processes. San Francisco, Calif.: Holden- Commun., tobe published; also in IBM Corp., Yorktown Day, 1962. Heights, N. Y., Res. Rep. RC-5062, Oct. 1974. 1291 J. W. Cohen, The Single Server Queue. New York: Wiley, 1969. [2] N. Abramson, “The ALOHA system-another alternative for 1301 R. Howard, DynamicProbabilistic Systems, Vol. 1: Markov computer communications,” in 1970 Fall Joint Comput. Conf., Modelsand Vol. 2: Semi-Markov and Decision Processes. New AFIPS Conf.Proc., vol. 37. Montvale, N.J.: AFIPS Press, York: Wiley, 1971. 1970, pp. 281-285. (311 E. J. Craig, Laplace andFourier Transforms for Electrical [3] W. Crowther, R. Rettberg, D. Walden, S. Ornstein, and Engineers. New York: Holt, Rinehart, and Winston, 1964. IEEE TRANSACTIONSON COMMUNICATIONS,VOL. COM-~~,NO. 4, APRIL 1975 423

Leonard Kleinrock (S’55-M’64-SM’71-F’73) data compressipn, priority queueing theory, and theoretical studies was born in New York, N. Y., on June 13, of time-shared systems. 1934. He received theB.E.E. degreefrom Dr. Kleinrock is a member of Tau Beta Pi, Eta Kappa Nu,Sigma theCity College of New York, N. Y., in Xi, the Operations Research Society of America, and theAssociation 1957, andthe S.M.E.E. andPh.D. degrees for Computing Machinery. He was awarded a Guggenheim Fellow- inelectrical engineering from the Massa- ship in 1971. chusettsInstitute of Technology, Cam- bridge, in 1959 and 1963, respectively, while participatinginthe Lincoln Laboratory Staff Associate Program. * From 1951 to 1957, he was employed at the PhotobellCompany, Inc., New York, N. Y., anindustrial electronics firm. He spent the summers from 1957 to 1961 at the M.I.T. Lincoln Laboratory, Lexington, Mass., first in the Digital Simon S. Lam (S’69-M174) was born in Computer Group and laterin the Systems Analysis Group. At M.I.T. Macao on July 31, 1947. He received he was a Research Assistant, initially with the Electronic Systems the B.S.E.E. degree in electrical engineering Laboratory, and laterwith the Research Laboratory for Electronics, from Washington State University, Pullman, where he worked on communication nets in the Information Process- in 1969, and the M.S. and Ph.D. degrees in ing and Transmission Group. After completing his graduate work at engineering from the University of California, the end of 1962, he worked at Lincoln Laboratory on communication Los Angeies, in 1970 and 1974, respectively. nets and on signal detection. In 1963 he accepted a position on the At the University of California, Los Ange- faculty atthe University of California, Los Angeles, where he les, he held a Phi Kappa PhiFellowship from is now Professor of Computer Science. He is a referee for numerous 1969 to 1970, and a Chancellor’s Teaching scholarly publications, book reviewer for several publishers, Fellowship from 1969 to 1973. He also par- and a consultant forvarious aerospace, research, and govern- ticiwated in the ARPA Network wroiect at UCLA as a Dostmaduate mental organizations. He is principal investigator of a large contract research engineer from 1972 to -197”4 and did research^ onsatellite with the Advanced Research Projects Agency (ARPA) of the De- packet communication. Since June 1974 he has been a research staff partment of Defense. He has published over 60 papers and is the member with the IBM Thomas J. Watson Research Center, York- author of Communication Nets; Stochastic Message Flow and Delay town Heights, N. Y. His current research interests include com- (New York: McGraw-Hill, 1964), Queueing Systems, VoZ. 1: Theory puter-communication networks and queueing theory. and Vol. B: Computer Applications (New York: Wiley-Interscience, Dr. Lam is a member of Tau Beta Pi, Sigma Tau, Phi KappaPhi, 1975). His main interests arein communication nets, computer nets, Pi Mu Epsilon, and the Association for Computing Machinery.

Quantization Error in Predictive Coders

Abstract-Predictive coders havebeen suggested foruse asanalog the theoretical rate distortion function for the first-order Markov data compression devices. Exact expressions for reconstructed signal process by about 0.6 bits/sample at low bit rates. error have been rare in the literature. Infact most results reported in the literature ire based on the assumption of Gaussian statistics for prediction error. Predictive coding of first-order GaussianMarkov I. INTRODUCTION sequences are considered in this paper. A numerical iteration technique is used to solve for the prediction error statistics expressed HE PREDICTIVE codershown in Fig. 1 has been as an infinite series in terms of Hermite polynomials. Several inter- suggested for video and voice coding applications. The esting properties of predictive coding are thereby demonstrated. T First, prediction error is in fact close to Gaussian, even for thebinary usefulness of predictive coding for data compression and quantizer. Sencond, quantizer levelsmay beoptimized at each itera- digitization of analog signals is well known, yet dueto its tion according to the calculated density. Finally, the existence of nonlinear nature, few exact solutionsfor quantization error correlation between successive quantizeroutputs isshown. Using the can be found. Let us note that the signal yk can be con- series solutions described above, performance in terms of mean- sidered a sample functionof a Gaussian Markov sequence square reconstruction error versus bit rate can beshown to parallel generated accordingto therecursion equation

Paper approved bythe Associate Editor forCommunication N Theory of the IEEE Communications Society for publication after yk = wk + C anyk--n (1) presentation at the1972 Information Theory Symposium, Asilomar, n=l Calif. Manuscript received August 6, 1974; revised October 9, 1974. This work was supported by the U.S. Army under Grant DA-ARO- and Wk is a sequence of independent unit variance (zero D-31-124-71-G89 through a research assistantship at the Depart- ment of System Science, University of California, Los Angeles, mean) Gaussian variables. Such functions are known as Calif. Theauthor iswith theComputer Sciences Corporation,Falls Gaussian autoregressive sequences and may be used to Church, Va. 22046. model signals whose spectra contain nozeros.