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Analysis of Queuing Behavior

of Automatic ESM Systems

M. H. EL-AYADI

Ain Shams University

Egypt

K. EL-BARBARY

H. E. ABOU-BAKR

Military Technical College

Egypt

Radar electronic support measures (ESM) systems detect

active emitters in a given area and determine their identities and

bearings. The high arrival rate of radar pulses in dense emitterenvironments demands fast automatic processing of arriving

pulses so that the ESM system can fulfill its functions properly

in real time. Yet, the performance analysis of automatic ESM

system in real life is difficult since both pulse arrivals and widths

can be specified only probabilistically. The success of queuing

theory in many applications such as computer communication

networks and flow-control has encouraged designers to utilize

queuing theory in qualifying and judging the performance of

automatic ESM systems in dense emitter environments. The

queuing behavior of these systems is analytically evaluated under

different service disciplines and elaborate computer simulations

validate the results. The analysis involves statistical modeling of

arrival and departure processes as well as distribution of service

times. It permits estimating the blocking probability due to high

arrival rates of intercepted radar pulses or due to limited speed of

the deinterleaver processor. Queuing analysis is shown to be quite

useful to quantitatively assess tradeoffs in ESM systems design.

Manuscript received February 1, 2000; revised May 31 and

November 2, 2000 and January 4, 2001; released for publication

February 9, 2001.

IEEE Log No. T-AES/37/3/08569.

Refereeing of this contribution was handled by J. P. Y. Lee.

Authors current addresses: M. H. El-Ayadi, 101 El Alamein Street,

El Sohfaeyeen, Cairo, Egypt; K. El-Barbary, Electrical Engineering

Department, Military Technical College, Cairo, Egypt; H. E.

Abou-Bakr, Electrical Engineering Department, Ottawa University,

Kingston, Canada.

0018-9251/01/$10.00 c2001 IEEE

I. INTRODUCTION

Early radar electronic support measures (ESM)systems relied on human operator interpretation ofESM receiver output to provide classification andidentification of intercepted emitters. The steadilyincreasing density of radar pulse environments leadsto the requirement of some form of automatic ESMprocessing to cope with high radar pulse arrival rates

and to provide a real-time response. AutomaticESM systems consist of three main subsystems:1) the receiver-encoder subsystem, that measures theparameters of each received radar pulse and encodesthem by a digital word called pulse descriptor vector(PDV), 2) the deinterleaver (or the preprocessor)that rapidly sorts the PDVs into sequences eachcomprising a group of PDVs supposed to be emittedfrom the same radar, and 3) the main processor.The deinterleaver much reduces the data rate sinceit encodes every segregated sequence of PDVs bya single emitter descriptor vector (EDV). The mainprocessor compares estimated EDVs with others

stored in the threat library of ESM system in orderto identify the type of intercepted radar. In someadvanced ESM systems, the main processor canfurther from identified radar type, instantaneousposition data supplied from onboard navigator,angle of arrival (AOA) information, and electronicorder of battle (EOB) [1] stored in the threat library,determine the location of the detected emitter. Thequeuing behavior of the receiver-encoder and thedeinterleaver in dense emitter environments isanalyzed here.

Queuing theory is concerned with the abstractmathematical modeling of systems subject to demandswhose occurrences and lengths can, in general, bespecified only probabilistically. Although, thesesystems are usually very complex, it is often possibleto abstract from the system description a mathematicalmodel whose analysis yields useful information aboutthe quality of the service and the efficient utilizationof the system. Here, reception of radar pulses, andextraction of their parameters as well as sorting ofPDVs into separate chains is modeled as a finite-statemachine operating as a queue with a single server. Itis known that the application of the queuing theory ispossible in principle if both the arrival processes at the

queuing model input and the service time distributioninside it are statistically characterized. All queuingmodels discussed here are conventionally labeled asshown in Fig. 1 [4, 5].

Fig. 1. Queuing model labels.

1010 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 37, NO. 3 JULY 2001

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TABLE I

Relation Between Arrival Rate of Radar Pulses at ESM Receiver

Input and Sum of PRFs of Radars Illuminating ESM Receiver

=

NXi=1

PRFiNumber of measured

Active Radars [Pulse/s] [Pulse/s]

7 31485 31500

8 33951 34000

9 35441 35500

10 38687 38750

11 42439 42500

12 47681 47750

13 49388 49500

14 52188 52250

15 56362 56500

16 58621 58750

17 59857 60000

18 60867 61000

19 62860 63000

II. QUEUING MODEL REPRESENTATION OF ESMRECEIVER-ENCODER SUBSYSTEM

A. Distribution of Interarrival Times

In dense emitter environment, the interarrivaltime between successive pulses at the ESM receiverinput is a random variable distributed according toa negative exponential distribution with parameter equal to the sum of the pulse repetition frequencies(PRFs) of all active radars in the instantaneous viewof ESM receiver [6]. Since for negative exponentialdistribution 1= is the mean value (of interarrivaltime), then is also the average arrival rate of radarpulses. As the number and PRF diversity of activepulse emitters increase, the arrival process at theinput of ESM receiver tends to be stationary Poissonprocess. That is given a time interval , the probabilitythat exactlyn pulses arrive at the ESM receiver inputduring is

Pn() = e()

n

n! : (1)

The average arrival rate was measured throughcomputer simulations of different number ofactive emitters with different PRFs.1 Both thecalculated arrival rate and the measured onemeasuredare provided in Table I. Clearly, the arrivalrate of radar pulses at the ESM receiver input isdirectly proportional to the number of active radarsilluminating the ESM receiver, and is equal to the sumof PRFs of these radars. The values of the interarrivaltime between successive radar pulses, intercepted inone observation interval of the ESM system wererecorded. The moments of the interarrival time up tothe fourth order were computed and denoted as Mim ;1 im 4. The first four moments, Mit ; 1 it 4 ofthe negative exponential distribution with parameter

1Details of simulation of instantaneous view of an ESM system are

provided in Appendix A.

taken to be the sum of PRFs of simulated active radarswere calculated. We defined a normalized squareddistanceD2 between the theoretical and the measureddensities of the interarrival time as

D2 =

M1t M1mM2t M2mM3t M3mM4t

M4m

2,

M1t

M2t

M3t

M4t

2

(2)

where,kxk denotes the norm of the vector x.Small value ofD 2 indicates closeness of measureddistribution to the theoretical one. Thus, a smallervalue ofD 2 indicates a better fit. Even though smallis good, the question how good a fit is given by aparticular value ofD2 should be answered. Therefore,we assumed that the measured interarrival timecould be expressed as a convex mixture of a randomvariate with the theoretical distribution and a normalrandom variate N(0,1), accounting for the simulationand modeling imperfections. The mixture weightswere for the random variate with the theoretical

distribution and (1) for the normal variate, where0 1. Thus, could be interpreted as a puritypercentage. CalculatingD 2 for the convex mixtureas function of and in (29), we could establishcorrespondence between the values ofD 2 and thoseof for given . This correspondence helped usinterpret the significance of the values ofD2, obtainedfrom computer simulations. The results are presentedin Table II for different simulated arrival rates ofradar pulses emitted from the active radars in theinstantaneous view of ESM receiver. For the datain Table II, 0:26106 D2 5:90106 whichcorresponds to 0:99979

0:99988.

The third property, we verified, is that thenumber of received pulses at the ESM receiver inputduring any observation time is randomly distributedaccording to Poisson distribution with parameter .We simulated an instantaneous field of view with = 30000 pulse/s and observation time = 0:21 s.2

Then, we partitioned this observation interval into anumber of subintervals N. We counted the number ofpulsesn occurring in each subinterval. We determinedthe frequencies of occurrence of different values ofn.We calculated the least squares estimate (cL) where

L = 0:21=Nand assuming a Poisson distributionmodel of measured frequencies. We applied the

chi-square goodness-of-fit test [10]. We repeatedthe above procedure for different lengths L of thesubintervals as shown in Table III. For the three casesconsidered in Table III, the least computed P -valueis 0.1025. Hence, we would not reject, at the = 0:1level of significance, the hypothesis that the number

2The value 0.21 s was the largest observation time we could

simulate for ESM system on the machine Pentium 166 MHz, given

= 30000 pulse/s.

EL-AYADI ET AL.: ANALYSIS OF QUEUING BEHAVIOR OF AUTOMATIC ESM SYSTEMS 1011

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TABLE II

Normalized Square Distance Between Theoretical Negative Exponential Distribution and Measured Distribution of Interarrival Times

Between Received Radar Pulses at Different Arrival Rates

measured M1m M1t M2m M2t M3m M3t M4m M4t D2

[pulse/s] [105] [105] [109] [109] [1013] [1013] [1017] [1017] [106]

31485 3.17 3.17 2.09 2.05 2.10 1.90 2.80 2.40 2.20

33951 2.90 2.90 1.90 1.70 1.90 1.70 2.60 1.70 2.06

35441 2.80 2.80 1.86 1.50 1.89 1.34 2.50 1.50 0.39

38687 2.50 2.50 1.50 1.33 1.50 1.03 1.30 1.00 0.26

42439 2.30 2.30 1.30 1.01 1.16 0.78 1.40 0.73 2.0547681 2.09 2.09 1.07 0.87 0.83 0.55 0.99 0.46 2.00

49388 2.02 2.02 0.98 0.82 0.81 0.45 0.95 0.39 3.30

52188 1.95 1.95 0.73 0.73 0.79 0.42 0.90 0.32 1.65

56362 1.77 1.76 0.82 0.63 0.66 0.33 0.81 0.23 5.90

58621 1.70 1.75 0.78 0.57 0.63 0.29 0.78 0.20 4.70

59857 1.67 1.66 0.76 0.55 0.63 0.27 0.76 0.18 2.89

60867 1.64 1.63 0.75 0.53 0.61 0.26 0.75 0.17 4.70

62860 1.59 1.58 0.72 0.51 0.60 0.23 0.72 0.15 4.30

67590 1.47 1.47 0.62 0.43 0.51 0.19 0.65 0.11 5.54

of pulses, received during given observation times,follows a Poisson distribution.

B. Distribution of Service TimesThe distribution of service time inside the

ESM receiver depends on what pulse parametersare measured and on the service discipline of thereceiver-encoder subsystem [1, 2]. Generally, there aretwo service disciplines of an ESM receiver-encoder[1, 2], namely the paralyzable counter withconstant dead time and the nonparalyzable counter.Both service disciplines do not allow waiting.Under the paralyzable service discipline, an ESMreceiver-encoder [1, 2] can only process pulse whicharrives after fixed time from the previous received

pulse. The corresponding queuing model is theM=D=1=0 model3 [4, 5]. Typical values of fixedservice time are the mean and the maximum widthsof arriving pulses. Under the nonparalyzable counterservice discipline an ESM receiver-encoder is ready toprocess a new coming pulse as soon as the previouspulse is expired. Thus the service time for eachreceived pulse is equal to its width. For overlappingpulses the service time is the minimum of resultantwidth and certain maximum permissible value [8].This avoids overloading by a CW signal.

The distribution of the service time underparalyzable counter service discipline is given by

(t ) and under the nonparalyzable counterdiscipline is given by the actual distribution of thewidths of arriving radar pulses. A good statisticalmodel of the service time is the Erlang distribution

3TheM=D=1=kmodel assumes that all service times have actuallyequal fixed values (deterministic values) and that we have a Poisson

input process with fixed mean arrival rate, only one server and

maximum number of customers allowed to wait is k. The symbolMindicates that the interarrival time is exponentially distributed,

while the symbol D stands for a deterministic service pattern.

with parameters and [4, 5],

f(t) = ()t1et

! (3)

Eftg=1

(4)

=q

Eft2gE2ftg = 1p

(5)

Eftg = 1p

: (6)

Notice that for = 1, the Erlang distributionreduces to the exponential distribution. This wouldbe the distribution of the service time under thenonparalyzable service discipline when the maincontributors to the arrival process at the input ofESM system are short pulse radars (like range-findersor tracking radars). For > 1, the mode of Erlangdistribution shifts rightward from zero and thecoefficient of variation =Eftg decreases. This mightapproximate the distribution of the service time underthe nonparalyzable counter service discipline in denseenvironment of surveillance radars with low diversityof pulsewidths. Ideally, the distribution of widths ofarriving pulses should be a multimodal distributionwith the number of modes corresponding to thenumber of active pulse radar emitters. But assuminglow diversity of widths and considering multipath andnarrowband filtering effects the actual distribution

might be a unimodal distribution. Finally, as !1,the Erlang distribution tends to the deterministicone (t1=). Thus, the queuing model of anESM receiver under nonparalyzable counter servicediscipline is M=G=1=0,4 [4, 5].

4TheM=G=1=0 assumes that the queuing system has a singleserver, maximum number of customers allowed to wait is zero,

and a Poisson input process (exponential interarrival times). The

customers are assumed to have independent service times but with

the same (general) probability distribution.


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TABLE III

Chi-Square Goodness-of-Fit Test for Number of Received Pulses

at ESM Receiver Input During Given Observation Times

N= 2100, L= 100 s

Lease squares (bL) = 2:9669j Interval Nj Npj

1 f0, 1g 430 428.712 f2g 479 475.65

3 f3g 500 470.404 f4g 347 348.915 f5,6, : : :g 344 376.33

s=P

(NjNPj)2=Npj= 4 :68Degrees of freedom = 511

P[2 > s] = 0:1969

N= 1050, L= 200 s


1 f0,1,2,3g 159 178.542 f4g 160 149.933

f5

g 184 173.90

4 f6g 165 168.085 f7g 158 139.246 f8,9,10, : : :g 224 240.31

s=P


P[2 > s] = 0:1311

N= 700, L= 300 s


1 f0,1, : : : , 6g 145 159.222 f7, 8g 193 180.163

f9,10

g 188 172.75

4 f11,12,13, : : :g 174 187.88s=P


P[2 > s] = 0:1025

III. EVALUATION OF FREE STATE PROBABILITYPf(t)

Under either service disciplines the ESMreceiver-encoder is in one of the following two states.

1) Free State: where the ESM receiver-encoder isfree and ready to receive any new coming pulse. The

probability that the ESM receiver is free is denoted byPf(t).2) Busy State: where the system is busy

processing a pulse and cannot receive and processanother pulse. So, in this state the new coming pulsewill be lost. The probability that the receiver is busy isdenoted by Pb(t).

Since the system must be in one of the above states, itis evident that

Pf(t) + Pb(t) = 1: (7)

The ESM receiver-encoder will be in the free stateat time instant t + t if either of the events e1 or e2happens.

e1: The ESM receiver-encoder is free at the instant tand no pulse arrives during the interval t.e2: The ESM receiver-encoder is busy at time instantt, processing of current radar pulse terminates duringthe subsequent interval t and no new pulse arrives.

From (1) we have

Pn=0(t) = e t (8)

P(e1) = Pf(t)e t Pf(t)(1 t): (9)

In [9], it is shown that the probability P (e2) can beexpressed in the form of0t, no matter the servicediscipline adopted and the distribution of the servicetime. Table IV provides expressions for 0 in fourdifferent cases. Therefore, we can express Pf(t + t)as

Pf(t + t) = Pf(t)(1 t) + 0t (10)which reduces to a first-order differential equation

d

dtPf(t) + Pf(t) =

0: (11)

The steady state solution of the above equation is

Pf(t) = 0=: (12)

Substituting from Table IV into (10) we get thefollowing cases.

Case 1Pf(t) = e

(13a)

Case 2

Pf(t) = 11 + w (13b)

Case 3

Pf(t) =

1 +

w

(13c)

Case 4

Pf(t) =ewmin ewmax

(wmaxwmin) : (13d)

Fig. 2 compares the ratios of successfullyprocessed pulses under paralyzable service discipline(case e) and under nonparalyzable service discipline(cases a, b, c, and d). Note that M =E

1=1=0

M=D=1=0 with= 1=. We assume that successfullyprocessed pulses are only those completely received.Fig. 2 indicates that for a given arrival rate, as theparameter of the Erlang distribution increases,the ratio of successfully processed pulses undernonparalyzable service discipline decreases. But, forall finite it remains higher than the same ratio underparalyzable service discipline. In practice, there canexist dense emitter environments providing pulsearrival rates much higher than those presented in


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Fig. 2. Comparison of ratios of successfully processed pulses

under paralyzable and nonparalyzable counter service disciplines,

PW is distributed according to Erlang distribution with

w= 1= = 12 micro seconds and for different : (a) = 1,(b) = 2, (c) = 5, (d) = 8, (e) = 1 (paralyzable).

Fig. 2. Clearly, at such very high arrival rates the ratioof successfully processed pulses will be extremelylow. Possible cures are either to equip the ESMsystem with several receiving-encoding channels,(multiple-servers), or to limit the measurements ofmonopulse parameters to those that can be obtainedfrom the pulse leading edge and imposing theparalyzable service discipline. In this case, the fixedservice time is of order 0:1s or even 0:05s. From(13a), the ESM receiver-encoder will be able toprocess up to 106 pulse/s with success ratio 90%.The same conclusion can be derived from Fig. 2.The coordinates of the first point on curve e are:

1= 31485 and success ratio r1= 65:56%, given= 12 s. Clearly, exp(010) = [exp(1)]c withc = 01

0=1. Thus, for 1= 106 and0= 0:1s, we

have r 01= r0:26471 = 89:43%.

IV. ESM RECEIVER-ENCODER DEPARTURE PROCESS

Under the paralyzable counter service discipline,the departure process is a renewal process since

TABLE IV

Expressions for 0 in Four Different Cases

Service Discipline Service Time Distribution Queuing Model 0

Case 1 Paralyzable Fixed value M=D=1=0 e

Case 2 Non- par alyzable Exponential with mean value = w(1) M=M =1=0

1 + w

Case 3 Non- paralyzable Erlang with par ameter and mean w(1) M=E=1=0

1 +

w

Case 4 Non-paralyzable Uniform in [wmin, wmax]

(2) M=G=1=0 ewmin ewmax(wmax wmin)

Note: (1)w is mean width of arriving pulses.(2)wmin and wmax are maximum and minimum widths of arriving pulses, respectively.

the pulses of the input flow which are blockedare not randomly selected. The departure processhas therefore a residual effect and is not a Poissonprocess. The distribution of the interdeparture timesof successive processed pulses fTd(l) is deduced fromthe interarrival times of the input flow of pulses asfollows

fTd(l) l=

fthe proportion of the distribution of theinterarrival times in the range (l, l + l)

gfthe proportion of the distribution of theinterarrival times in excess ofg

=el l

e , l > (14)

fTd(l) = exp((l )); l (15)

where Td is the interdeparture time between outputPDVs from the receiver-encoder subsystem and isthe fixed service time of the receiver. We evaluatedthe squared distance between the theoretical andthe measured densities of the interdeparture times,according to (2). The results are given in Table Vfor different simulated arrival rates, assuming =11 s. For the data in Table V, 0:29104 D2 6:57104. Using (30), we interpret this result as0:99911 0:99956.

Now we are going to show that undernonparalyzable counter service discipline if the arrivalprocess is a Poisson process, then so is the departureprocess.Yet it is necessary that the width of intercepted

pulses be exponentially distributed or has an Erlang

distribution with moderate .Suppose that n pulses arrive at the input of

the ESM receiver during L seconds and only d

of them are successfully processed by the ESMreceiver-encoder subsystem and passed to thedeinterleaver processor in the form ofd PDVs. Thus,nd pulses arent processed or are missed. Asindicated in the previous section each arriving pulsewill with probabilityPffind the ESM receiver in freestate and with probability Pb in busy state. Hence,the number d of pulses successfully processed outofn pulses arriving during the interval L follows the


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TABLE V

Normalized Square Distance Between Theoretical Delayed Negative Exponential Distribution With Parameters and and MeasuredDistribution of Interdeparture Times Between Output PDVs at Different Arrival Rates


[pulse/s] [105] [105] [109] [109] [1013] [1013] [1017] [1017] [104]

31485 4.30 4.27 2.50 2.25 2.73 2.70 3.49 3.44 0.49

33951 4.03 4.04 2.36 2.15 2.02 2.21 2.68 2.60 0.66

35441 3.88 3.90 2.33 2.13 2.10 1.90 2.58 2.40 1.09

38687 3.70 3.68 2.24 2.02 1.74 1.50 1.75 1.60 0.29

42439 3.46 3.45 2.06 1.74 1.52 1.24 1.27 1.17 0.8447681 3.25 3.19 1.78 1.46 1.27 0.93 1.19 0.78 0.48

49388 3.20 3.12 1.72 1.38 1.29 0.85 1.08 0.69 6.57

52188 3.05 3.01 1.71 1.27 0.92 0.79 1.04 0.57 1.12

56362 2.90 2.85 1.56 1.19 0.85 0.62 1.01 0.44 3.07

58621 2.84 2.80 1.02 1.07 0.80 0.61 0.97 0.38 2.04

59857 2.90 2.77 1.38 1.04 1.13 0.53 0.88 0.36 1.28

62860 2.85 2.65 1.30 0.97 1.05 0.47 0.75 0.31 3.50

Note: Constant service time = 11 s.

TABLE VI

Normalized Square Distance Between Theoretical Negative Exponential Distribution With Parameter 0 and Measured Distribution ofInterdeparture Times Between Output PDVs at Different Arrival Rates


[pulse/s] [105] [105] [109] [109] [1013] [1013] [1017] [1017] [104]

31485 4.07 4.17 2.50 3.40 2.55 4.33 3.31 7.20 5.90

33951 4.01 3.90 2.55 3.11 2.50 3.68 3.20 5.80 3.16

35441 4.02 3.80 2.50 2.90 2.50 3.30 3.20 5.10 2.40

38687 3.60 3.50 2.10 2.50 1.90 2.70 2.40 3.90 0.74

42439 3.30 3.33 1.79 2.25 1.50 2.26 1.50 3.17 0.05

47681 3.05 3.09 1.45 1.91 1.14 1.71 1.30 2.20 5.80

49388 2.99 3.02 1.39 1.83 1.09 1.66 1.20 2.17 0.93

52188 2.88 2.95 1.30 1.70 1.05 1.49 1.20 1.70 11.00

56362 2.69 2.70 1.18 1.50 0.93 1.28 1.10 1.40 8.00

58621 2.65 2.70 1.14 1.46 0.90 1.18 1.10 1.20 9.00

59857 2.63 2.67 1.20 1.42 0.90 1.14 0.95 1.10 0.31

60867 2.62 2.64 1.16 1.39 0.90 1.10 0.92 1.10 0.02

62860 2.60 2.59 1.14 1.34 0.90 1.04 1.00 1.08 0.82

67590 2.45 2.47 1.02 1.22 0.82 0.91 0.95 0.90 0.65

Note: Nonparalyzable counter service discipline, average PW = 10 s.

binomial distribution,

Pd=n(L) =

n

d

PdfP

ndb , 0 d n (16)

where Pd=n(L) is the probability thatd PDVs willemerge from the ESM receiver-encoder subsystemgiven that n pulses are received at its input duringtime L. Let us recall that in dense environments thenumber of pulses arriving at the ESM receiver inputis a random variable distributed according to Poissondistribution with parameter . The probability Pd(L)thatd PDVs will emerge from the receiver-encoder in

L seconds regardless of the number of arriving pulses,can be expressed using the law of total probability;i.e.,

Pd(L) =1X

n=d

Pd=n(L)Pn(L) =1X

n=d

n

d

PkfP

ndb Pn(L):

(17)

Substitution of (1) into (17) gives

Pd(L) =(0L)

d

d! e0L, d 0 (18)

where0= Pf: (19)

From (18) it is clear that under the assumptionsindicated above the departure process will be aPoisson process with departure rate 0= Pf.Consequently, the distribution of the times betweensuccessive output PDVs from the receiver-encodersubsystem is a negative exponential distribution withparameter0= Pf.

Computer simulations verified these conclusionsand the results are presented in Table VI, where0:017104 D2 11104, corresponding to0:99920 0:99985.

If the assumption made on the distribution ofwidths of arriving pulses is absent, i.e., if for example


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0 < wmin PW wmax, then the values ofd in(16)(18) should be restricted to the following ranges:in (16), [L=wmax] d minf[L=wmin], ng, and in(17), [L,wmax] d [L=wmin], where [x] denotes theintegral part ofx. Obviously in this case the departureprocess is no longer a Poisson process since (18) isnot valid for all d, L 0.

V. APPLYING QUEUING THEORY TOMULTIPLE-PARAMETER DEINTERLEAVING

Computer-based deinterleaving algorithmsare classified as interval-only algorithms andmultiple-parameter deinterleaving algorithms.Interval-only algorithms operate on theinterpulse-arrival times to estimate potential pulserepetition interval (PRI) values corresponding to thestream of PDVs coming out of the receiver-encodersubsystem. Every potential PRI value is used tosegregate a repetitive sequence of PDVs, assumedto represent a single radar emitter. A well-known

interval-only algorithm using sequential difference(SDIF) histograms is given in [7]. In dense emitterenvironment, (high pulse arrival rate and large numberof interleaved radar pulse sequences), interval-onlyalgorithms are of low efficiency and reliability.Multiple-parameter deinterleaving algorithmsoperate on the monopulse parameters. Usually, theAOA and the radio frequency are used to sort thePDVs into strings with similar primary parameters.Multiple-parameter deinterleaving is completedby PRI analysis of every segregated string, usingSDIF histogram [7]. It is possible to use the pulseamplitude (PA) for later antenna-scan analysis, but

this costs the deinterleaver a large amount of datastorage.

We focus here on the queuing behavior ofmultiple-parameter deinterleavers. In [1, p. 452],deinterleaving is defined as memory-intensiveprocedure converting PDVs into EDVs. A typicaldeinterleaving strategy for modern ESM system isdescribed in [1, pp. 55, 56]. In Fig. 5 we presenta possible implementation of this strategy. First, afrequency band coarse presort of arriving PDVs isdone. Actually, each PDV is labeled with a frequencyband number and is directed (through a demultiplexer)to one ofN paths according to its frequency band

label (Nis the total number of frequency bands). Ineach path, a fine RF sort of arriving PDVs is carriedout by attaching to every PDV a label indicating theserial number of the frequency bin comprising itsRF. Suppose that every frequency band comprises Mfrequency bins. After the fine RF sort, a histogramof active RFs is constructed by counting the numberof PDVs belonging to each one of the Mfrequencybins. PDVs, already served by the histogram of activeRFs, are stored in a content addressable memory

(CAM). The histogram of active RFs compares thecount in every frequency bin with two thresholds.Once the count in any frequency bin exceeds theupper threshold, the presence of stable high-dutyfactor emitters (CW radars, pulsed-Doppler radars,and data-links) is declared. The complete label ofthis particular frequency bin is sent to the CAM toretrieve all PDVs with this label. They are sortedon the AOA, and the resulting strings are passed

to PRI analysis. Simultaneously, the upper andlower limits of the frequency bin in which the countexceeded the high threshold are sent to an adaptivestatic filter (employing window addressable memory(WAM)) to block all new PDVs from the detectedhigh duty factor emitters. This allows subsequentprocessing to be executed at a reduced data rate.After completing the histogram of active RFs, thelabel of every frequency bin with final count fallingbetween the two thresholds is used to identify thePDVs, already stored in the CAM, and are belongingto frequency stable emitters. Again these PDVs aresorted on the AOA and the resulting strings are sent

to PRI analysis. At last, the labels of the remainingfrequency bins with final count below the lowerthreshold are used to retrieve from the CAM all thePDVs that are supposed to belong to frequency-agileemitters. They are grouped in one string, sorted onAOA, and then passed to PRI analysis. In all cases thenonagile components of the PDVs of every segregatedsequence are averaged to encode the whole sequenceby a single EDV.

Let us now show how to apply queuing theoryto the multiple parameter deinterleaving processdiscussed above. We start with the coarse presort. Lettcomp be the time needed to compare the frequency

of a new arriving PDV with the upper and lowerlimits of one of the Nfrequency bands. If a sequentialsearch is used, then at most N comparisons are neededto identify the band that matches the PDV beingserved. Hence, the service time in the coarse presortis a discrete random variable with probability densityfunction (pdf) of the form

bs(t) =NXi=1

p(i)(t i:tcomp) (20)

where p(i) is the probability that the incoming PDVmatches the ith frequency band,PNi=1p(i) = 1. If thefrequency bands are ordered and a binary search isused, then at most log2N comparisons are needed.The service time is again a discrete random variablewith pdf,

bb(t) =

log2NXi=1

p0(i)(t i:tcomp) (21)

where p0(i) is the probability that the match is exactlyfound after i comparisons,

Plog2Ni=1 p

0(i) = 1. If the


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coarse presort is realized by employing a WAM, thefrequency component of the new arriving PDV issimultaneously compared with the upper and lowerlimits of all frequency bands. Hence, the pdf of theservice time will be

bw(t) = (t tcomp): (22)If no high duty factor emitters have been detected,the adaptive static filter is not introduced in the

path of the main stream of PDVs departing fromthe receiver-encoder. Then, with the latter under theparalyzable counter service discipline, the queuingmodel of coarse presort will be fTd=b=1=0, where

fTd is given by (15) and the density b(t) is given by(20), (21), or (22). But with the receiver-encoderunder the nonparalyzable counter service disciplineand assuming exponentially distributed widths ofintercepted pulses, the coarse presort queuing modelwill be M=b=1=0 with arrival rate 0= =(1 + w),(from (13b) and (19)). In this case, expressions(23a)(23c) of blocking probabilities correspondrespectively to the densities (20), (21), and (22) ofthe service time in the coarse presort.

Sequential search

pb= 1NXi=1

p(i)e0itcomp (23a)

Binary search

pb= 1[log2N]X

i=1

p0(i)e0i:tcomp (23b)

WAM

pb= 1 e0:tcomp

: (23c)In order to estimate the primary speed requirementof the coarse presort, let us consider the followingexample: Pb= 10%,N= 8, 0= 30000 PDV/s,

p(i) = 1=Nfor 1 iN, p0(i) = 1=[log2N] for 1i [log2N] and 4 instructions=tcomp (2 subtractions tocompare with RF upper and lower limits, 1 fetch, 1load). With sequential search the processor will haveto run at an instruction rate r 5:054 MIPS, withbinary searchr 2:258 MIPS, and with WAM r 1:139 MIPS. IfNis increased to 16, the instructionrate should be raised above 9.528 MIPS for sequential

search and above 2.817 MIPS for binary search. Butfor WAM, the instruction rate need not be increased.Once high duty-factor emitters are detected, the

adaptive static filter is introduced before the coarsepresort to censor all their subsequent PDVs fromthe main flow. It is necessary then to modify thedistribution of interarrival times of PDVs at thecoarse presort. Assume that both the number ofdetected high duty-factors emitters and their sum ofPRFs,s, are small compared with the number andsum of PRFs of other emitters. Assume also that

tcomp< < Tmin where is the fixed service time ofthe receiver-encoder under the paralyzable counterservice discipline and Tmin is the minimum of PRIsof detected high duty-factor emitters. It can be shownthat the distribution of interarrival times of PDVs atthe coarse presort can be approximated either to

fI(l) = ( s)e(s)(l) l (24)with the receiver-encoder under the paralyzable

counter service discipline, or to

fI(l) = s1 + w

e[(s)=(1+w)](ltcomp ) l tcomp(25)

with the receiver-encoder under the nonparalyzablecounter service discipline and assuming exponentiallydistributed widths of intercepted pulses.

Partitioning the flow of PDVs after the coarsepresort intoNsubflows much alleviate the queuingproblems in the subsequent processing phases. Forexample, the pdf of the service time in the fine RFsort is given by (20), (21), or (22) with Nreplaced

by M, (the number of frequency bins), in (20) and(21) and according that a sequential search, a binarysearch or a WAM is implemented. Assume, just forinformative purposes, that expressions (23a)(23c)apply to the fine RF sort.5 Assume also that thecoarse presort blocks 10% of the arriving PDVs andthose who come out are equally distributed amongtheNavailable paths. Thus, if the arrival rate atthe coarse presort is = 30000 PDV/s, then for

N= 8 we will expect an arrival rate at the fine RFsort, = 30000 0:9=8 = 3375 PDV/s. Now, for

M= 16 and for the lowest instruction rate calculatedabove, (1.139 MIPS), we would expect Pb= 0:094 for

sequential search,Pb= 0:029 for binary search, andPb= 0:012 for WAM. Evidently, for higher instructionrates we have even smaller blocking probabilities.Furthermore, the service time in the histogram ofactive RFs is constant for all PDVs and is almostequal to 0:5tcomp which implies less blocking of PDVsthan in preceding stages. For the AOA sort as wellas for the PRI analysis the arrival pattern changesto batch arrivals at slow rate instead of single PDVarrivals at high rate. Thus, the AOA sort will almostcause no blocking of PDVs. The sizes of PDV batches(strings) arriving for PRI analysis are smaller thanthose of PDV batches arriving for AOA sort. Yet

the service time for each string in the PRI analysismay vary significantly, because the emitters maybe randomly agile in PRI and/or some PDVs in thestring may be missing due to blocking in previousprocessing phases.

From the above investigation of the queuingbehavior, it has become clear that the frequency band

5The distribution of interarrival times of PDVs at the fine RF sort is

no longer a negative exponential distribution.


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coarse presort is the true bottleneck of the studieddeinterleaving process. The performance of thedeinterleaver in terms of the blocking probability isimproved when it is preceded by a prebuffer of size

K. The function of this prebuffer is to store up to Kof the arriving PDVs when the coarse presort is busy.However, if a new PDV arrives while the prebuffer isfull, this PDV will be lost. The queuing model of thecoarse presort in this case is M=b=1=K, assuming the

receiver-encoder is under the nonparalyzable counterservice discipline.

Suppose that the deinterleaver is preceded by aprebuffer of sizeK. The instant at which the coarsepresort completes servicing the nth PDV is theinstant when the deinterleaver is ready to service the(n + 1)st PDV. Let us designate such instant by tn, n =1,2,3, : : : , and the corresponding state of the prebufferby the positive integer k, where kis the number ofPDVs left in the prebuffer after the last processedPDV. Now we denote the probability that the prebufferis at instant tn in statekbyk(tn), 0 kK. Assumethat the input flow of PDVs is a Poisson process with

average rate 0. The probability that jPDVs arriveat the prebuffer input during the service time of PDVnumbern is

rj= P(n=j) =

Z10

P(n=j, t) dt

=

Z10

e0t(0t)j

j! b(t) dt (26)

where n is a random variable denoting the number ofPDVs arriving at the prebuffer input while servicingthe PDV number n and b(t) is given by (20), (21), or(22). The probabilitiesrjare transition probabilities

associated with occurrence of different prebuffer statesduring servicing successive PDVs. It can easily bededuced that

j(tn) =

jXi=[j=k]

ji+1(tn1)rj, j= 0,1,2, : : : ,K

(27)

where [x] denotes the integral part ofx. As n !1,the system composed of the prebuffer and the coarsepresort reaches a steady state and the prebuffer stateprobabilitiesk, 0 kKbecome time independent.In particular, the steady state blocking probability willbe given by the limit ofk(tn) as n !1. Moreover,the system (27) will take the form

jXi=[j=k]

ji+1(i,1) = 0j,0, j= 0,1,2, : : : ,K

(28)

where m,n equals 1 ifm = n and equals zero form 6= n. The above system can be efficiently solvedfor 1,2,3, : : : ,Kin terms of0, using forward

Fig. 3. Dependence of coarse presort blocking probability on the

arrival rate of PDVs, constant prebuffer size = 5 PDVs,

comparison time with one frequency band limits: (a) 45 s,(b) 35 s, (c) 25 s, (d) 15 s. N is total number of active

emitters within instantaneous view.

substitution. By imposing the constraint,PK

i=0 i= 1we can determine the value of0 and hence the values

of the other state probabilities i, 1 i K. However,any analytic expression for the steady state blockingprobability will be in expanded form that does noteasily provide useful conclusions.

VI. COMPUTER SIMULATIONS AND CONCLUSIONS

We simulated a receiver-encoder under thenonparalyzable counter service discipline, followedby a prebuffer of size Kand a coarse presort into Nfrequency bands using sequential search. Figs. 3 and4 show quantitatively the effect of the speed of the

coarse presort processor and the size of the prebufferon the blocking probability of the coarse presort atdifferent arrival rates of PDVs. Fig. 3 shows thatfor a fixed prebuffer size and fixed average arrivalrate of PDVs, the higher the speed of the presortprocessor, the lower the blocking probability. Also,the blocking probability is reduced by increasingthe size of the prebuffer, for fixed processor speedas shown in Fig. 4. In particular, for the simulatedemitter environment, when the single comparisontime with RF limits of one frequency band is 35sand the prebuffer size is 5 PDVs, then for arrivalrates

20000 PDV/s the blocking probability is

50% (Fig. 3). Now, if the prebuffer size is increasedfrom zero to 80 PDVs, the presort can process upto 23400 PDVs/s with blocking probability 50%(Fig. 4). Alternatively, if the single comparison timeis decreased from 35 s to 25 s (by increasing theprocessor speed), the coarse presort with prebuffersize 5 PDVs can process up to 24000 PDVs/s withblocking probability 50% (Fig. 3). Thus in ourcase, 40% increase of processor speed is roughlyequivalent to 16 times increase of the prebuffer size


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Fig. 4. Dependence of coarse presort blocking probability on the arrival rate of PDVs, constant comparison time with one frequency

band limits = 35 s, (a) prebuffer size = 0 PDVs, (b) prebuffer size = 20 PDVs, (c) prebuffer size = 40 PDVs, (d) prebuffer size = 80

PDVs. N is total number of active emitters within instantaneous view.

Fig. 5. Typical multiple-parameter deinterleaving scheme.

if the blocking probability is the measure ofcomparison. We could also assess another interestingtradeoff in design of coarse presort for simulatedemitter environment. Fig. 4 indicates that at tcomp=35 s,0= 27650 PDV/s, N= 8, and prebuffer size

K= 80, the blocking probability is estimated to be0.62. Meanwhile, expression (23c) indicates that thesame value of blocking probability will be obtainedwhen employing a WAM for coarse presort andremoving the prebuffer.

In this paper it has been shown that queuingmodels closely match experimental data and that

queuing analysis is quite useful to quantitativelyassess tradeoffs in ESM systems design. For example,the number of bits used to generate the PDV willdetermine the needed comparison time inside thedeinterleaver. Thus, the longer the length of thePDV, the higher the blocking probability of thedeinterleaver. To decrease the comparison time, thenumber of bits used to represent the PDV shouldbe decreased, but this degrades the resolution andaccuracy of measuring the parameters of interceptedpulses. To maintain good measurement performanceof each parameter without increasing the blocking

probability of the deinterleaver we have either toincrease the prebuffer size or to choose a fasterprocessor to decrease the comparison time. Although,fast processors are expensive they ensure high averageservices rates, minimal reporting latency of interceptedthreats and are usually associated with less hardware(no need for large buffers). The service time of thedeinterleaver can also be decreased by optimizingthe sorting algorithms and in some applications, bylimiting the measurement of monopulse parameters inthe ESM receiver-encoder only to those obtained fromthe pulse leading edge (short PDVs).

Finally, the authors suggest the followingextensions of this work: 1) development andevaluation of techniques for adaptation of servicediscipline to emitter environment; 2) further queuinganalysis of design tradeoffs: single sever with limitedwaiting room (buffer) versus multiple servers, singleparameter deinterleaver versus multiple parameterdeinterleaver, high resolution of monopulseparameter measurements versus deinterleaver errorperformance; 3) queuing analysis of electroniccounter measures (ECM) resource manager and


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in particular the effect of different priorities ofdemands.

APPENDIX A

A. Simulation of Instantaneous View of ESM System

We assume that the radar emitters existent inthe instantaneous view of an ESM system have

technical parameters in the following ranges: PRF[16] KHz, PW [110] s, and RF [1.512.5] GHz.The AOAs of intercepted pulses vary from [571].

Simulated radars are of low probability of intercept

(LPI) and therefore have very low sidelobe level(SLL40 dB). Thus the ESM receiver interceptsonly mainlobe emissions from these radars. As we aresimulating instantaneous views of the ESM system,

the observation intervals are much shorter than thescan cycle of any of the simulated radars. Therefore,

the individual radar scan rates are not consideredin the simulations. The mainlobes of intercepted

radars illuminate continuously the antenna of the

ESM system during the relatively short observationinterval (chosen to be 0.075 s). Thus, for an average

pulse arrival rate of the order of 40000 pulse/s,the ESM receiver is expected to intercept about

3000 pulses/observation interval which is convenientfor the storage capacity and the speed of the machine

used for simulation (Pentium 166 MHz). Eachintercepted pulse is represented by a PDV with

components: RF, PW, AOA, and time of arrival ofthe pulse. We consider also both frequency-stable

and frequency-agile radars. In every simulationfrequency-agile radars are randomly selected from

the set of active radars in the instantaneous viewof ESM system. They are assigned agility bands

falling completely within the frequency range ofthe instantaneous view of the ESM system (agilityband = 510% of the RF). The pulses received from

all radars during the observation time are sorted withrespect to their time of arrival in ascending order.

This results in a stream of randomly interleaved radarpulses simulating an instantaneous view of the ESM

system.

B. Expressions of Squared DistancesD 2 for ConvexMixtures

1) The theoretical distribution isf(l) = exp(l1), l 0

D2(, ) =

(1 )4 + 92

2(1)4 +h

12

22(1)2 +3(1 )4

i222

+ 44

4+ 36

6

6 +576

8

8

:

(29)2) The theoretical distribution isf() =

exp((l )), l

D2(, , ) =

(1)4 + 92(1)4

1

+

2+

62(1)2

2

2+

2

+ 2

+3(1)4

22

1

+

2+ 4

2

2+

2

+ 2

2+ 6

6

3+

6

2 +

32

+ 3

2+ 8

24

4 +

24

3 +

122

2 +

43

+ 4

2 :(30)

REFERENCES

[1] Curtis, S. (1986)

Introduction to Electronic Warfare.

Dedham, MA: Artech House, 1986.

[2] Davies, C. L., and Hollands, P. (1982)

Automatic processing for ESM.

IEE Proceedings, Pt. F, Comm., Radar & Signal Process,

129, 3 (1982), 146171.

[3] Tsui, J. B. (1986)

Microwave Receivers with Electronic Warfare Applications.

New York: Wiley, 1986.

[4] Hillier, S., and Lieberman, J. (1995)

Introduction to Operation Research(6th ed.).

New York: McGraw-Hill, 1995.

[5] Bunday, D. (1986)Basic Queuing Theory.

London: Arnold, 1986.

[6] Bussgang, J. J., and Fine, T. L. (1963)

Interpulse interval distribution in the environment of N

periodic pulse radars.

IEEE Transactions on Radio Frequency Interference,RFI-5

(1963), 710.

[7] Milojevic, D. J., and Provic, B. M. (1992)

Improved algorithm for deinterleaving of radar pulses.

IEE Proceedings, Pt. F, Comm., Radar & Signal

Processing, 139, 1 (1992), 98104.

[8] Wilkinson, D. R., and Watson, A. W. (1985)

Use of metric techniques in the ESM data processing.

IEE Proceedings, Pt. F, Comm., Radar & Signal

Processing, 132, 7 (1985), 221225.[9] Abou-Bakr, H. (1998)

Analysis of the queuing behavior of automatic ESM

system in dense emitter environments.

Master of Science thesis in Electrical Engineering,

Military Training College, Cairo, Egypt, 1998.

[10] Low, A. M., and Kelton, W. D. (1991)

Simulation, Modeling & Analysis (Industrial Engineering

Series).

New York: McGraw-Hill, 1991.


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M. H. El-Ayadi received the B.S. degree (with distinction and honor) in electricalengineering from the Military Technical College, Egypt, the M.S. degree inelectrical engineering from the VAAZ Academy, Czechoslovakia, the Ph.D.degree in signal processing and automatics from the University of Paris XI,France, in 1969, 1976, and 1980, respectively.

He was a faculty member at the Military Technical College, Egypt inthe Radar Engineering Dept. (19801981), then in the Electronic WarfareEngineering Dept. (19821996). There he became a Professor of Electrical

Engineering (1991), Head of Electronic Warfare Engineering Dept. (1992),and Head of Electrical Engineering Depts. (1995). Since 1996, he has been aProfessor of Signal & Systems at Ain-Shams University, Egypt, in the ComputerSystems Dept. He supervised the inauguration of the Scientific Computing Dept.(19982000). His research interests include statistical signal and array processingand related applications in radar, communications, electronic warfare, distributeddetection, and data fusion.

Khairy El-Barbarywas born in Cairo, Egypt on November 22, 1958. Hereceived the B.Sc. and M.Sc. degrees in electrical engineering from the MilitaryTechnical College, Cairo in 1981 and 1986, respectively. He received thePh.D. degree in electrical engineering from the George Washington University,Washington, DC in 1994.

Since 1994 he has been a staff member of the Electrical EngineeringDepartment in the Military Technical College, Cairo, Egypt. His technicalinterests lie in the area of adaptive signal processing, interference cancellation,and array processing with applications to communications and radar systems.

Hussam Abou-Bakrwas born in Cairo, Egypt on January 26, 1971. He receivedthe B.Sc. and M.Sc. in electrical engineering from the Military Technical College,Cairo, Egypt in 1992 and 1998, respectively.

During the period 1993 to 1999 he was a teaching assistant in the Departmentof Electrical Engineering, Military Technical College, Cairo. Currently he ispreparing for the Ph.D. degree on electrical engineering at the Ottawa University,Kingston, Canada. He is interested in the area of signal processing with radarapplications.


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