Tandem behavior of a telecommunication system with finite buffers and repeated calls

Queueing Systems, 6 (1990) 89-108 89

T A N D E M B E H A V I O R O F A T E L E C O M M U N I C A T I O N S Y S T E M W I T H F I N I T E B U F F E R S A N D R E P E A T E D C A L L S

Behnam POURBABAI

Department of Mechanical Engineering, The University of Maryland, College Park, MD 20742, USA

Received 16 June 1988; revised 10 October 1989

Abst rac t

The tandem behavior of a telecommunication system with finite buffers and repeated calls is modeled by the performance of a finite capacity G/M/1 queueing system with general interarrival time distribution, exponentially distributed service time, the first-come-first-served queueing discipline and retrials. In this system a fraction of the units which on arrival at a node of the system find it busy, may retry to be processed, by merging with the incoming arrival units in that node, after a fixed delay time. The performance of this system in steady state ~s modeled by a queueing network and is approximated by a recursive algorithm based on the isolation method. The approximation outcomes are compared against those from a simulation study. Our numerical results indicate that in steady state the non-renewal superposition arrival process, the non-renewal overflow process, and the non-renewal departure process of the above system can be approximated with compatible renewal processes.

Keywords: Queueing network, approximation, retrials, telecommunication, simulation, overflow process, departure process, superposition process.

1. Introduction

Cons ider a t e l ecommunica t ion system with the fol lowing character is t ics : Mes- sages have to be t ransmi t ted by a single server device (e.g., a processor) with a finite capac i ty buffer . If the system is full at the ins tants of arr ival of the messages at the system, some of the messages will no t be t r ansmi t t ed ( that is, collisions occur) , bu t a f ract ion of the original messages will be r e t r ansmi t t ed af te r a f ixed de lay t ime (see fig. 1). The new messages are the over f low (coll ided) units which are no t initially t ransmi t ted b y the t ransmit ter . In this sys tem a f rac t ion of the messages which have not ini t ial ly been t ransmi t t ed will be al lowed to leave the sys tem and not be t ransmit ted . F o r a discussion and an appl ica t ion of this system, see R io rdan [19] and Pourbaba i [13].

The p r i m a r y object ives of this pape r are to (1) ap p ro x im a te the p e r f o r m a n c e o f the above t e l ecommunica t ion system, and (2) a p p r o x i m a t e its t a n d e m behav io r in

�9 J.C. Baltzer A.G. Scientific Publishing Company

arrival

process

/ buffer

\ f ~ overflow

process

server

2

superposition ~

overflow

arrival process~.~rocess

Fig.

1. A

sys

tem

wit

h th

ree

fini

te c

apac

ity

G/M

/1 q

ueue

ing

syst

ems

wit

h re

tria

ls.

B. Pourbabai / Tandem behavior of a telecommunication system 91

steady state. For this purpose, a queueing network model is developed and analyzed by a recursive algorithm. The behavior of each node of the network is modeled and analyzed by the steady state performance of a compatible single server finite capacity G/M/1 queueing system with generally distributed interarrival time, exponentially distributed service time, the first-come-first-served queueing discipline, and retrials (repeated calls).

In the queueing literature, three methodologies can be distinguished for the steady state analysis of networks. We call them the product form method, the decomposition method, and the isolation method. All of these methods can be used for the exact and the approximation analysis of some queueing networks. In all these methods, the basic idea is to characterize the arrival process at each node of the network and then to evaluate the appropriate performance measures at the same node. More specifically, in the product form method, the primary focus is on characterizing the distribution of the number of arrivals at each node of the network; in the decomposition method, the primary focus is on obtaining the parameters of the arrival rate and the coefficient of variation (c.v.) of the distribution of the interarrival time at each node of the network; and in the isolation method, the primary focus is on characterizing the distribution of the interarrival time at each node of the queueing network. The limitations and capabilities of these methods differ from one another, but we shall not discuss them here. However, we note that the decomposition method and the isolation method have many similarities and one major difference. Thus, while in both methods the departure process of each node of the network should be approximated, in the decomposition method only the c.v. of the interarrival time distribution at each node is approximated without using the actual distribution, whereas in the isolation method the interarrival time distribution of each node is obtained. Hence the advantage of the isolation method is that, if necessary, more than the first two moments of the interarrival time distribution at each node can be derived from the corresponding distribution, which in some cases can improve the accuracy of the corresponding approximation algorithm. Moreover, the isolation method can be applied for both open and mixed queueing networks, and if necessary, the role of the overflow process of each finite capacity node can also be explicitly considered. For a review of the literature of the queueing networks emphasizing on the product form method, see Sauer and Chandy [20], and the references cited there. For a comprehensive presentation of the decomposition method, see Kuehn [5] and Whitt [22]. For a review of the basic idea of the isolation method and its applications, see pourbabai and Sonderman [8,9] and Pourbabai [10-16].

In this paper we develop a non-Markovian queueing network for performance modeling of a telecommunication system consisting of a set of single server devices, each with a finite capacity buffer and repeated calls. For this network we developed an algorithm for analyzing both the performance and the tandem behavior of a finite capacity G/M/1 queueing system with retrials, generally

92 B. Pourbabai / Tandem behavior of a telecommunication system

distributed interarrival and service times and demonstrate that in steady state the non-renewal overflow process, the non-renewal superposition arrival process, and the non-renewal departure process of each finite capacity G / M / 1 queueing system with retrials can be approximated with compatible renewal processes.

The organization of this paper is as follows. In section 2, the main results are presented. In section 3, numerical results are presented along with some concluding remarks.

2. The main model

To develop the proposed algorithm in this paper the following assumptions are made. (1) The non-renewal overflow process and the non-renewal departure process of

each node (e.g., each finite capacity G/M/1 queueing system with retrials) of the network can be approximated with compatible (e.g., phase type) renewal processes.

(2) The thinned overflow processes of each node can be approximated with compatible renewal processes.

(3) The superposition arrival process of each node can be approximated with a compatible renewal process. That is, the dependencies among the respective interarrival, the interoverflow, and the interdeparture times are ignored, and compatible phase type distributions are fitted to the corresponding distributions.

Throughout the L: i: j:

ka and ca:

k a and ca:

~:~j and c~j:

~.Oj and c~

~O/j and c'~:

paper the following notations will be used. number of nodes; the node index; the superposition index which counts the number of times the overflow process superimposes with the arrival process; the arrival rate and the c.v. of the distribution of the interarrival time; the arrival rate and the c.v. of the distribution of the interarrival time of the arrival process at the node i; the superposition arrival rate and the c.v. of the distribution of the interarrival time of the superposition arrival process at the node i and at the iteration j ; the overflow rate and the c.v. o f the distribution of the inter - overflow time of the total loss units at the node i and at the iteration j; the overflow rate and the c.v. of the distribution of the inter - overflow time of those units which will superimpose with the arrival process at the node i and at the iteration j ;


~0j and E~ the overflow rate and the c.v. of the distribution of the interoverflow time of those units which will permanently leave the system at the node i and at the iteration j ;

/~ and Ki: the processing rate and the buffer capacity of the node i; Li}(t), L~(s) and (-1)mL/~m(O): the distribution of the interoverflow time, the

Laplace-Stieltjes transform of the distribution of the interoverflow time at the node i and at the iteration j , and its ruth moment;

Dij(t), DiT(s ) and ( - 1)mDi~m(o): the distribution of the interdeparture time, the

qi:

Laplace-Stieltjes transform of the distribution of the interdeparture time at the node i and at the iteration j , and its ruth moment; the stationary probability that an arbitrary unit which initially has not been processed at the node i will attempt to retry to be processed.

cO = xo {Ly,(o)_

Hence,

x0 = - [ L y ( o ) ] i j

where o o

L~.(s) = fo e-St dLij(t)"

Similarly, the parameters of the departure process can be expressed.

(1)

(2)

3. The algorithm

To approximate the performance of the system, the following steps will be implemented:

Step 0: set i = 1 and j = 1. Step 1: at iteration j > 0, given that at least the values of the first two

moments of the distribution of the interarrival time of the node i (e.g., G / M / 1 / K i queueing system) have been identified, a compatible distribution will be fitted to those moments, and the dependencies among the interarrival times of the superposition arrival process will be ignored. That is, the nonrenewal superposition arrival process will be approximated bY a compatible (e.g., phase-type) renewal process. In other words, the performance of the ith G / M / 1 / K i queueing system will be approximated by the performance of a compatible G 1 / M / 1 / K i queueing system with a renewal input in steady state, see appendix A.

Step 2: at iteration j > 0, the overflow process of the G I / M / 1 / K i queueing system will be approximated based on the results of Akulinichev and Ivannikov [1] and Pourbabai [11]; also see appendix B. For this purpose, at least the first


two moments of the distribution of the interoverflow time of the G I / M / 1 / K ~ queueing system will be obtained, and based on these moments the parameters (X~ cO.) of the overflow process of the node i and at iteration j will be obtained.

Step 3: the overflow processes corresponding to the overflow units which will either permanently leave the system or retry to be processed, will be approximated. For this purpose, the overflow process of the queueing system will be thinned using the results of Gnedenko and Kovalenko [3] on the thinning of a renewal process. The parameters (~0/y, ?o.) and (~0./, 3oy) of the thinned overflow processes are obtained as follows:

~oy= q~XOj, cij = [ ( 1 - q~)+ qi(ciO)2] 1/2 (3)

and

" ,,0 [ ~/ 0 x211/2 )•/j= (1 - qi)X~ cij= qi + (1 - qi)(ciy) J �9 (4)

Step 4: a fraction of the overflow units at iteration j > 0 which will retry to be processed will be superimposed with the incoming arrival units at node i. It is noted that as orginally indicated by Pritsker [17] and later confirmed by Nawijn ([7], p. 170), the recirculation time (that is, the length of time it takes a rejected message to superimpose with the incoming arrival messages) at each node will not influence the performance of the system in steady state. The resulting superposition arrival process will be approximated using the asymptotic method of Whitt [21] for approximating a finite number of stationary counting processes. Thus

X,+, (5) X~/'J+' = ~,.j + X~, (6)

e,+, (7) s a a2 1/2 (xiA+ ) x',,j+, (8)

Step 5: after approximating the parameters of (h~,j.+~, c~,j+~), the previous steps (e.g., steps 1 to 5) will be repeated until a steady state is reached at iteration "e" . The steady state is identified as follows. Let 8 be a sufficiently small value (e.g., 1 • 10-3), and

then

m = m i n ( j : X s + + l - x',+.< n=min(j : c~,j+,-ciT <~ 8); (9)

e = max(m, n). (10)

Step 6: in steady state, the departure process of the G I / M / 1 / K i queueing system with retrials will be approximated using the results of Laslett [6] and


Pourbabai [8]; also see appendix B. For this purpose, at least the first two moments of the distribution of the interdeparture time of the G I / M / 1 / K i queueing system will be obtained. Based on these moments the parameters ~0.. c~e) of the departure process of the node i at iteration j will be obtained, ie~

and a compatible (e.g., phase-type) distribution will be fitted to those parameters; that is, the departure process will be approximated with a compatible (e.g., phase-type) renewal process.

Read L / ~ , set i71 and j=l

<--NO

�9 j , c i j ) ~ ~ the "~verflow ro~cesses oT the G/M/I/K i

q u e u e i n g system w i t h hyperexponentially distributed �9 " times. i nterarrival

,, I ~ ~ YES

I Approximate (~~ ~~ and (%oij , COl j) of the overflow processes of the G/M/I/K i q u e u e i n g system w i t h hypoexponentially distributed Interarrival times. .

Approximate (~sij, cSij)

Steady St;te?

YES

NO L

Fig. 2. Flowchart of the approximation algorithm.

96 B. Pourbabai / Tandem behavior o f a telecommunication system

Step 7: after reaching a steady state at iteration "e" , the performance of the i th queueing system can be evaluated based on its utilization factor and other performance measures such as the net departure rate, the overflow rate, and the superposition arrival rate of the node i. Denote

d Pie = h i e / ~ i " (11)

Step 8: if > L, stop. Else, set

i = i + 1, h a = hdi_l,r c a = c d (12) i-- l ,e

and go to step 1. For a summary of the approximation steps, see fig. 2.

4. Numerical results and concluding remarks

In this section we present examples for L = 3, qi = 0.5, /~i = 1, and i = 1 to 3, and compare the approximation outcomes against those from a simulation study. The parameters of hie and Pie are approximated based on the propositions 2 and 3 of appendix B. To investigate the accuracy of the approximation outcomes, a simulation model was also developed, using the SLAM simulation package developed by Pritsker and Pegden [18]. Each simulation outcome was obtained based on 80,000 departing units. The approximation and the simulation outcomes are presented in figs. 3-8. Note that because the service rates are equal to one, the utilization factor of each node is equal-to the departure rate in steady state. Moreover, from figs. 3-8, the overflow rate of each node can be obtained by finding the difference between the superposition arrival rate and the departure rate.

From figs. 3-8, the following conclusions are drawn for the observed range of values. (1) The approximation algorithm is relatively accurate. (2) For each value of the c.v. of the distribution of the interarrival time, as the

arrival rate increases, the superposition arrival rate and the utilization factor increase.

(3) For each value of the arrival rate, as the c.v. of the distribution of the interarrival time increases, the superposition arrival rate of the first node increases, but the superposition arrival rate of the other two nodes and the utilization factor of the other nodes decrease.

As can be observed in section 3, the p roposed algorithm is relatively accurate. However, the most interesting result from our analysis is that ignoring the dependencies among the interarrival times, the interoverflow times, and the interdeparture times of each finite capacity G / M / 1 queueing system, and approximating their corresponding distributions with compatible phase-type distributions, have not significantly influenced the accuracy of the proposed algorithm in steady state.

I 0 0 0 0 0 0 0 0 0 0

0 9 8 7 6 5 4 3 2 I

It

J

0

02

Simulation

ca=.8

Approximation

ca=.8

Simulation

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Approximation

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Simulation

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Approximation

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I I

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ca=.8

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Simulation

ca=.8

Approximation

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Simulation

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Approximation

c a=l-0

Simulation

ca=l.5

Approximation

c a=l-5

0 0 0 1

0 0

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0.2

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The unique feature of our model is that the performances of similar telecommunication systems even with the deterministic arrival process can also be analyzed. For this purpose, the arrival process can be modelled by a renewal process with shifted exponentially distributed interarrival time, while the c.v. of the corresponding distribution can be set to be sufficiently close to zero. To improve the accuracy of the proposed algorithm, we suggest the use of more than the first two moments of the interoverflow time, the interdeparture time, and the interarrival time distributions.

Finally, the tandem behavior of a random access telecommunication system can also be approximated. For this purpose, the performance of the latter telecommunication system can be modelled by a G / M / N / K queueing system with N parallel processors, a buffer of size K, the random access processing discipline, and retrials. However, because the departure process of this queueing system is not known, given that N is sufficiently large (e.g., more than 5), it is recommended to approximate the departure process with a compatible Poisson process. Then the proposed algorithm can be applied as described in section 2.

A p p e n d i x A

As discussed in Kuehn [5] and Whitt [21], to approximate the distribution of the interarrival time of a stationary arrival process with the arrival rate 2t and the c.v. of the distribution of the interarrival time c >1 1, the following hyperexponen- tial distribution function with two parameters and balanced means can be used:

n 2 ( o , 7; t) --- 1 - 0 e -v~t- (1 - 0) e -Y2t, t >/0, (1.1)

where the shape parameter is

cZ +----~ + 1 //2 (1.2)

and the intensity parameters are

a = 2 O X , � 8 9 (1.3)

When 1 / v ~ - < c < 1, the following hypoexponential distribution function with two parameters can be used:

Ez(N; t) 1 a2 e -~'' al - = - - - - e ~=t, t >~ 0 , ( A . 4 ) Ot 2 - - 0t 1 O/1 - - O~ 2

where the intensity parameters are

a 1 = 2X/[1 + (2c 2 - 1)1/z], ot 2 = 22k/[1 --(2c 2 - 1)1/2]. (1.5)


Appendix B

Let the service times (Sk} be independent and identically distributed with Pr( S k ~< t} = B( t ) , t >1 0 and for all k. Potential units arrive at successive epochs of a stationary arrival process with interarrival times ( T k } such that Pr{ T k < t } = A( t ) , A(0 + ) = 0. An arrival finding a queue with K units waiting, does not enter the queueing system, and is considered a lost unit (that is, such a unit is not considered as part of the departure process). The kth served arrival who finds n~, units in the queueing system (queue and service), waits for a time W k until service and is served for a time Sk. Denote by ( ~rj, 0 ~< d ~< K + 1 } the stationary state probabilities of the number of customers in the queueing system at the instant before an arrival occurs. These probabilities are known to exist. Let

rrj = erj , 0 ~<j < K. (B.1) Pr(n~ = j ) = % + . . . +Trx 1 - ~rr+ 1

For a G I / M / 1 / K queueing system with a renewal input,

Pr(Wk < ~ t ) = B n k * ( t ) , t>~O, (B.2)

where

B~ = 1 ift>_-0 (B.3) = 0 if t < 0 ,

and BJ*(t) is the j-fold convolution of B ( t ) = 1 - e -t~t with itself. Furthermore, let v ( k ) be the index of the kth served arrival, so that

if n k f v ( k ) < K, (B.4)

v(k + 1) = / i n f 1" Y'~ Tj:~ S ' k - r if n k = K, (B.5) j=v(k)+l

where Sff_ r is the time elapsing from the v(k)th arrival to the completion of servicing of the customer being serviced. Since the service distribution function B ( t ) is exponential, the distribution of S~_ r is also the same exponential distribution function. Moreover, the time between arrivals of the k th and (k + 1)th units to be served is given by

T; = T,(k)+, + " " + T,(k+l). (B.6)

Let Ik+ a be the idle time for the server between the completion of the k th service and the beginning of the (k + 1)th service, so that

Ik+ 1 = max(0, T; - Wk -- Sk). (B.7)

Thus, the departure process from the queueing system could be defined via the sequence (D~ } of the interdeparture times given by

ok = + tk, (B.8)

where S k and Ik are independent random variables, because the arrival process and service times are independent.

B. Pourbabai / Tandem behavior of a telecommunication system

PROPOSITION 1 For a G I / M / 1 / K queueing system and t >/0,

Pr{D, ~< t} = { B(t)[1

+ E ~/f~ + -v ) dA(v) / ( 1 - ~ + 0 , j = O 0

105

(B.9)

where

~r/= ~rj, O < j ~ < K - 1 ,

A*(p,) = fo e - ~ t d a ( t ) '

K + I

qrj= ~ qrj+,_lbn, I < j < K + I , Uo = n ~ O

.OOe-~t(#t) j bL=Jo j! dA(t ) , 0 ~<j~< K + 1.

The above results lead to the following results:

, ~rx = r + rrK+l, ~'k= I _ A * ( / ~ )

I=0

[]

(B.lO)

(B.11)

')] Y'~ b. , (B.12) n = 0

(B.13)

PROPOSITION 2 Let

A( t l= H2(O, 7~ ; t )= l - O e-X,t- (1-O) e -x2t,

and

t>~0 (B.14)

B ( t ) = l - e -~t, t>~O. (B.]5)

Then, for an given by

-D*'(O) =

H J M / 1 / K queueing system, the first two moments of D/~ are

(1) [1-~o-~/~A*(.)]

ETrj 0 ~ 1 j = O ~'1 ~ ' ~ " ~kZ - - - -

j + l

+(1-0) X2_~.?t-~- A2+ # # Xz+/.t ) J/ / (1 - "/7"K+ 1 ) , (B.16)


(2) D*"(0)= 1 -%-r 772

+ E ~ ; o -~ 2 ),+1

}~i + tt " #2

2 h2 x~-3~- 7= + x~+--7 + ( 1 - 0 )

/(1 - ~rK+,),

where ~f is obtained from proposition 1, and

Ohl]s j (1 - - 0)~2~.s j bi= (?,,+~)j+, + (;~2+~)j+,, 0< j<K+a .

(B.17)

[] (B.18)

PROPOSITION 3 The Laplace-Stieltjes Transform of the distribution of the interoverflow time

of a G I / M / N / K queueing system is

L*(s) ~- PO*K+N(S) +

where

K+N E P* ( s ) Z ~ + N - i + I ( S ) ' ( B . 1 9 ) i,K+N

i = 1

= * Z * Z*(s) P~(s)Z*+I(s ) + E Pit (s) r_i+l(S) i=l

I < ~ r < K + N - 1 ,

(B.20)

P i * r ( s ) = ( ~ ) f o ~ [ 1 - e x p ( - i x x ) ] i e x p { - [ s + ( r - i ) t z ] x } dA(x)

O <~ i <~ r < N,

[oo( Nl~x ]* exp[- (N# + s)x] dA(x) ei*,N+r(S) = ao ~---~. ]

) i-1 r . Nr+l N - 1 E E ( -1 ) a+l _j),+~

P~*+i'N+'(s) = i-- 1 j=o b=0 (i

• fo~((i _j)b(xtz)b exp[-(N-j) l . tX] - exp[- ( N - i)/zx] }

X exp(-sx) dA(x)

(B.21)

l <~r<K,O<~i<r,

(B.22)

(B.23)

1 < i < N , 1 < r < k .

(B.24)


For N = 1, K = 0 and N + 1, K = 1, proposit ion 3 yields the following two results.

See Akulinichev and Gorskiy [1] and Pourbabai [11]. []

PROPOSITION 4 The first two moments of the distribution of the interoverflow time of an

H2/M/1/O queueing system are

11 L * " ( 0 ) = - L * ' ( 0 ) = A0'

where

x2A0 + 211(A1 - 11)

A~ (B.25)

0xl (1-0)x2 0xl (1-0)x2 = ~ + A 1 = + (B.26) A0 Xl+~ X ~ + ~ ' (x1+~)~ (x~+~)~'

0 (1 - 0 ) 20 2(1 - 0 ) 11 = ~'1 "~ X2 ' 1 2 = ~-z + X 2 [] (B.27)

PROPOSITION 5 The first two moments of the distribution of the interoverflow time of an

H2/M/1 /1 queueing system are

11(1 - A I ~ ) - L * ' ( 0 ) = _~ , , (B.28)

A~

where

L * " ( 0 ) = A~ + B3 At

91----- ~ ( A I ~ - - 1 - A 0 ) (B.29) At

B 2 = 12 - 2/~A2G*'(0 ) + 2BI(A 1 - - I 1 + ~A2) , (8.30)

B 3 = (1 - A o - btA1)(I 2 - 211B1), (B.31)

207~ 1 2(1 - 0)• 2 = + (8.32) A= (Xl+~)' (X2+~) 3

Ao, A1, I 1, and I z are the same as those of proposit ion 4. [] To obtain similar results for an E 2 / M / 1 / K queueing system with hypoex-

ponentially distributed interarrival time and K = 0 and 1, in proposit ions 4 and 5,

set 0 = X2/(~k 2 -- ~kl).

References

[1] N.M. Akulinichev and V.A. Ivannikov, Limit distributions of additive-type functions in a particular queucing problem, Izv. Akad. Nauk Tech. Kibernetika 1 (1974) 44-51.


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Res. 6 (1981) 563-470. [5] P.J. Kuehn, Approximate analysis of general queueing networks by decomposition, IEEE

Trans. Comm. COM-27 (1979) 113-126. [6] G.M. Laslett, Characterizing the finite capacity G I / M / 1 queue with renewal output, Manage-

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The Netherlands (1983). [8] B. Pourbabai and D. Sonderman, A recirculation system with random access, Europ. J. Oper.

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Oper. Res. 32 (1987) 340-352. [11] B. Pourbabai, Approximation of the overflow process from a G / M / N / K queueing system,

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retrials, Int. J. Syst. Sci. 18 (1987) 985-992. [13] B. Pourbabai, A random access telecommunication system, Electronische Information Sverar-

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Elektronische Information Sverarbeitung und Kybernetik (E.I.K.), J. Information Processing and Cybernetics (1989).

[17] A.A. Pritsker, Application of multichannel queueing results to the analysis of conveyor systems, J. Industr. Eng. 17 (1966) 14-21.

[18] A.A. Pritsker and C.D. Pegden, Introduction to Simulation and SLAM (Halsted Press, 1979). [19] J. Riordan, Stochastic Service Systems (Wiley, 1962). [20] C.H. Sauer and K.M. Chandy, Computer System Performance Modeling (Prentice Hall, 1981). [21] W. Whitt, Approximating a point process by a renewal process I: Two basic methods, Oper.

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Documents

Tandem behavior of a telecommunication system with finite buffers and repeated calls