Approximation for overflow moments of a multiservice link with trunk reservation

Performance Evaluation 43 (2001) 259–268

Approximation for overflow moments of a multiservicelink with trunk reservationq

Andreas Brandta,∗, Manfred Brandtba Wirtschaftswissenschaftliche Fakultät, Humboldt-Universität zu Berlin, Spandauer Strasse 1, D-10178 Berlin, Germany

b Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, D-14195 Berlin, Germany

Abstract

In this paper we propose an approximation for individual overflow moments of a multiservice link with differing arrivalrates, capacity requirements and mean holding times, where trunk reservation is used. The approximation is a generalizationof Roberts’ well-known approximation for individual blocking probabilities of a multiservice link to higher moments. It canbe computed very efficiently. The quality of the approximation for the second moment (variance) is comparable to Roberts’approximation for the individual blocking probabilities. Thus the results provide an efficient algorithm for computing thetwo moment characterization of the individual overflow streams and hence can be used for the design and analysis of circuitswitched alternate routing networks with trunk reservation links. © 2001 Elsevier Science B.V. All rights reserved.

Keywords:Loss system; Trunk reservation; Bandwidth; Link overflow moments

1. Introduction

Recent developments in communication systems have led to much interest in systems where traffic ofwidely differing characteristics is integrated together. In the absence of controls, larger bandwidth callscan experience a much higher blocking probability than low bandwidth calls [1,2]. Also by excessivealternative routing during periods of general overload, instability of the network and performance degra-dation may occur, cf. [3–5]. The trunk reservation strategy (TRS) is a very efficient and simple admissionstrategy for equalizing blocking probabilities or for limiting excessive alternative routing. Moreover, it isa simple fix for certain forms of instability in networks. Further, the TRS can be used to effect an almostcomplete prioritization of the different traffic streams, while still utilizing the full capacity of the link.Various authors have considered models with TRS, cf. [1–21] and the references therein.

In this paper we propose an approximation for the individual overflow moments of a multiservicelink with TRS, i.e. for the moments of the number of individual overflow calls simultaneously in asecondary link of infinite capacity, which can be computed very efficiently and can be considered as a

q This work was supported by a grant from the Deutsche Telekom.∗ Corresponding author. Tel.:+49-30-2093-5622; fax:+49-30-2093-5745.

E-mail address:[email protected] (A. Brandt).

0166-5316/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0166-5316(00)00044-4

260 A. Brandt, M. Brandt / Performance Evaluation 43 (2001) 259–268

generalization of Roberts’ well-known approximation [19] for the individual blocking probabilities tohigher moments. In particular, thus we obtain an efficient approximation for the variance and hence a twomoment characterization of the individual overflow traffics of a link with TRS. This new approximationallows to extend the well-known algorithms for computing performance measures in circuit switchedalternate routing networks based on a two moment characterization of traffic streams, cf. [22,23], tonetworks having links with TRS.

We consider a link of integer capacityC offered a finite numberm of traffic streams indexed in the setI := 1, . . . , m. Calls of typei ∈ I arrive as a Poisson stream of rateλi and have exponential holdingtimes with mean 1/µi . Each such call requires an integer capacitybi (bandwidth) and is accepted if andonly if, after admittance, the link has at leastC −Ci remaining resource units; otherwise the call gets lost.The thresholdsCi ∈ bi, . . . , C, i ∈ I , are usually referred to as trunk reservation parameters. (Often thequantitiesC − Ci , i ∈ I , are also called trunk reservation parameters, cf. [8].) Without loss of generalitywe may assume thatC = max(C1, . . . , Cm) and, because of possible rescaling, that the bandwidthsbi ,i ∈ I , are relatively prime. The overflow traffic of type-i calls is routed to a secondary link of infinitecapacity where the holding times are exponentially distributed with mean 1/µi too. Although for practicalpurposes usually only the first two moments of type-i overflow calls are used for characterizing overflowtraffic streams we derive the results for arbitrary moments since the ideas work for higher moments, too.(For the use of higher moments in applications, cf. [18].)

The paper is organized as follows. In Section 2, we give linear systems of equations for the partialfactorial moments of the individual overflow streams. As mentioned above, for network dimensioningtools, fast and numerically stable algorithms for the (first two) overflow moments are needed. However,this cannot be achieved for the mentioned linear systems of equations in case of a larger number oftraffic types and growing capacity. Thus, in Section 3, we propose a new approximation for the overflowmoments, which can be computed very fast and may be considered as a generalization of Roberts’well-known approximation [19] for individual blocking probabilities to higher moments. In Section 4,we give some numerical examples showing that the quality of the approximation for the second (and third)moment is comparable to the quality of Roberts’ approximation for the individual blocking probabilities.Further we discuss the results and give some conjecture and remarks for further research.

2. Linear systems of equations for the overflow moments

LetNi(t) andN ′i (t) denote the number of calls of typei in progress in the primary and secondary link at

time t , respectively. Then the state of the primary link is given by the vectorN(t) := (N1(t), . . . , Nm(t)),andS(t) := N1(t)b1 + · · · + Nm(t)bm is the total amount of resource utilized by the calls in the primarylink at timet . The set of allowed statesΩ of N(t) is determined by the bandwidthsbi and trunk reservationparametersCi , i ∈ I , as well as by the capacity constraintC := max(C1, . . . , Cm): Ω := n ∈ Zm

+ :(n, b) ≤ C, nibi ≤ Ci for i ∈ I , wheren := (n1, . . . , nm), b := (b1, . . . , bm) and(n, b) := n1b1 +· · ·+nmbm. Letp(n), n ∈ Ω, andq0(`), ` ∈ 0, . . . , C, be the equilibrium probabilities of the processesN(t) andS(t), respectively. The probability that an arriving type-i call is blocked (individual blockingprobability) is given by

pb,i =∑

n∈Ω:(n,b)>Ci−bi

p(n) =C∑

`=Ci−bi+1

q0(`), i ∈ I. (1)

A. Brandt, M. Brandt / Performance Evaluation 43 (2001) 259–268 261

In case of TRS a product-form solution for the equilibrium distributionp(n) does not exist as one-waytransitions are present in the state space destroying reversibility [24]. However, in principle the loss prob-abilities — and, more generally, the overflow moments — can be computed by solving the correspondingsteady-state equations numerically. However, this becomes intractable as the number of call types andthe link capacity grow, therefore approximations are of interest.

If there is no trunk reservation, i.e.Ci = C, i ∈ I , then thep(n) are given by

p(n) = G−1(Ω)∏i∈I

%ni

i

ni !, n ∈ Ω,

whereG(Ω) is the normalization constant and%i := λi/µi , i ∈ I . The utilization probabilities satisfythe following recursion and normalization condition, originally due to [19,25]

`q0(`) =∑i∈I

%ibiq0(` − bi), ` = 1, 2, . . . , C,

C∑`=0

q0(`) = 1, (2)

whereq0(`) := 0 for ` ∈ Z \ 0, . . . , C. More general recurrences have been derived for state depen-dent arrival intensities by several authors, cf. [5,26]. The recursion (2) yields an efficient algorithm ofcomplexity O(Cm) for theq0(`) and hence also for the blocking probabilities.

Without loss of generality we will deal with the overflow stream of type-1 calls in the following. Theprocess(N(t), N ′

1(t)), t ∈ R+, is an irreducible Markov process with state spaceΩ × Z+. Let p(n, n′1),

(n, n′1) ∈ Ω × Z+, denote its equilibrium distribution,

fk(n) :=∑

n′1∈Z+

k!

(n′

1k

)p(n, n′

1), n ∈ Ω, fk =∑n∈Ω

fk(n) (3)

thekth partial factorial moment ofN ′1(t)onN(t) = nand thekth factorial moment ofN ′

1(t), respectively.Fork = 0 we have

f0(n) = p(n), n ∈ Ω, f0 = 1. (4)

The state equations for thep(n, n′1) readλ1 +

∑i∈I\1

λiI(n, b) ≤ Ci − bi +∑i∈I

niµi + n′1µ1

p(n, n′1)

=∑i∈I

λiIni > 0I(n, b) ≤ Cip(n − ei, n′1) +

∑i∈I

(ni + 1)µiIn + ei ∈ Ωp(n + ei, n′1)

+λ1In′1 > 0I(n, b) > C1−b1p(n, n′

1 − 1) + (n′1+1)µ1p(n, n′

1 + 1), (n, n′1)∈Ω×Z+,

whereei , i ∈ I , denotes theith unit vector inRm. Multiplying these equations by

k!

(n′

1k

),


summing overn′1 ∈ Z+ and using(

j + 1k

)=(

j

k

)+(

j

k − 1

), j

(j

k

)− j

(j − 1

k

)= k

(j

k

),

we find(∑i∈I

λiI(n, b) ≤ Ci − bi +∑i∈I

niµi + kµ1

)fk(n)

=∑i∈I

λiI(n, b) ≤ Cifk(n − ei) +∑i∈I

(ni + 1)µifk(n + ei) + kλ1I(n, b) > C1 − b1fk−1(n),

n ∈ Ω, k = 0, 1, . . . , (5)

wherefk(n) := 0, n ∈ Zm \ Ω, k = 0, 1, . . . . Obviously, (5) even holds forn ∈ Zm. Summation overn ∈ Zm yields

fk = %1

∑n∈Ω:(n,b)>C1−b1

fk−1(n), k = 1, 2, . . . . (6)

In particular, fork = 1 we obtainf1 = %1pb,1, cf. (1). Fork = 0, Eq. (5) obviously reduces to theequilibrium equations forp(n), n ∈ Ω. The f0(n) = p(n) are uniquely determined by (5) and thenormalization conditionf0 = 1. For fixedk > 0 and givenfk−1(n), n ∈ Ω, the related to (5) coefficientmatrix for fk(n), n ∈ Ω, is strictly row-wise diagonally dominant. Therefore from the strong row sumcriterion, cf. [27, p. 576] it follows that thefk(n), n ∈ Ω, are the unique solution of the linear system ofequations (5). Thus, in view of (6), the factorial momentsfk can be determined by a recursive computationof thefj (n), n ∈ Ω, for j = 0, 1, . . . , k − 1, in principle. However, this approach becomes intractablefor a larger number of call types and growing link capacity, due to numerical complexity. For up to threecall types the algorithm was implemented for checking the accuracy of the proposed approximation givenin the next section; for more than three call types we simulated the overflow moments.

3. Approximation for the individual overflow moments

Let

qk(`) :=∑

n∈Ω:(n,b)=`

fk(n), ` = 0, 1, . . . , C, k = 0, 1, . . . (7)

be thekth partial factorial moment ofN ′1(t) on S(t) = ` andqk(`) := 0 for all the other integer values

of ` andk. From (7), (3), (4) and (6) we obtain

C∑`=0

qk(`) = fk = Ik = 0 + %1

C∑`=C1−b1+1

qk−1(`), k = 0, 1, 2, . . . . (8)

Little’s formula yields the identity

C∑`=0

`q0(`) =∑i∈I

(C∑

`=0

I` ≤ Ci − biλibiq0(`)

)1

µi

=C∑

`=0

(∑i∈I

I` ≤ Ci%ibiq0(` − bi)

). (9)


For a motivation of our approximation given below we consider the special case ofb1 = · · · = bm,µ1 = · · · = µm, i.e. of equal bandwidthsbi and equal mean holding times 1/µi . By summing the resultingsimplified equations (5) overn ∈ Zm with (n, b) = ` and using (7) after some algebra we find(

(` + b1)qk(` + b1) −∑i∈I

I` + b1 ≤ Ci%ib1qk(`)

)−(

`qk(`) −∑i∈I

I` ≤ Ci%ib1qk(` − b1)

)= kb1qk(`) − kI` > C1 − b1%1b1qk−1(`), ` ∈ Z, k = 0, 1, 2, . . . . (10)

Since the r.h.s. of (10) vanishes fork = 0, we obtain the following well-known recursion for the utilizationprobabilities, cf. [19]:

`q0(`) =∑i∈I

I` ≤ Ci%ibiq0(` − bi), ` = 1, 2, . . . , C,

C∑`=0

q0(`) = 1. (11)

Note that (2) (special case of complete sharing) coincides with (11), i.e. in both special cases (11) holds. Inthe general situation (11) holds in the mean only, cf. (9). The similarity of the two special cases encouragedRoberts [19] to propose the unique solutionq0(`), ` = 0, . . . , C, of (11) as an approximation for theutilization probabilitiesq0(`), ` = 0, . . . , C. In view of (1) this provides also an approximationpb,i ,i ∈ I for the individual blocking probabilities. Several authors have observed over the years that thisapproximation is accurate if the mean holding times do not greatly differ from each other, but that it istypically inaccurate otherwise [5]. Let us return to the special case ofb1 = · · · = bm, µ1 = · · · = µm inmore detail. For ∈ Z andk = 0, 1, 2, . . . , let

rk(`) :=∞∑

j=0

I` − jb1 ≥ 0qk(` − jb1), sk(`) :=∞∑

j=0

I` − jb1 > C1 − b1qk−1(` − jb1).

These sequences satisfy the recursions

rk(`) = qk(`) + rk(` − b1), sk(`) = I` > C1 − b1qk−1(`) + sk(` − b1), (12)

whererk(`) = sk(`) = 0, ` ∈ Z \ Z+. Using (12) from (10) we obtain

(` + b1)qk(` + b1) −∑i∈I

I` + b1 ≤ Ci%ib1qk(`) − kb1rk(`) + k%1b1sk(`)

= `qk(`) −∑i∈I

I` ≤ Ci%ib1qk(` − b1) − kb1rk(` − b1) + k%1b1sk(` − b1). (13)

This implies that the r.h.s. of (13) has periodb1. Thus the r.h.s. of (13) vanishes for` ∈ Z as it obviouslyvanishes for ∈ Z \ Z+. This together with (12) yields

`qk(`) =∑i∈I

I` ≤ Ci%ib1qk(` − b1) + kb1rk(` − b1) − k%1b1sk(` − b1)

=∑i∈I

I` ≤ Ci%ib1qk(` − b1) + kb1

∞∑j=1

I` − jb1 ≥ 0qk(` − jb1)

−k%1b1

∞∑j=1

I` − jb1 > C1 − b1qk−1(` − jb1), ` = 1, 2, . . . , C. (14)


Suppose that for a fixedk ∈ Z+ theqk−1(`), ` = 0, 1, . . . , C, are known. Then (14) and (8) determineuniquelyqk(`), ` = 0, 1, . . . , C. An efficient recursive algorithm of complexity O(C(m + k)/b1) forcomputingfk can be derived, which is similar to those for the approximation in the general case givenbelow, but of lower complexity.

In the general case of arbitrary bandwidths and mean holding times the following approximationqk(`)

of qk(`) is suggested: for = 1, 2, . . . , C andk = 0, 1, . . . , let

`qk(`) :=∑i∈I

Ibi ≤ ` ≤ Ci%ibi qk(` − bi)

+k(µ1∑

i∈I I` ≥ bi%ibi rk(` − bi) − λ1∑

i∈I I` ≥ bi%ibi sk(` − bi))∑

i∈I λi

, (15)

where for` = 0, 1, 2, . . . , C andk = 0, 1, . . .

rk(`) := qk(`) +∑

i∈I I` ≥ bi%ibi rk(` − bi)∑i∈I %ibi

, (16)

sk(`) := I` > C1 − b1qk−1(`) +∑

i∈I I` ≥ bi%ibi sk(` − bi)∑i∈I %ibi

. (17)

With q−1(`) := 0, ` = 0, 1, . . . , C, and the normalization condition corresponding to (8)

C∑`=0

qk(`) = Ik = 0 + %1

C∑`=C1−b1+1

qk−1(`), k = 0, 1, 2, . . . , (18)

the qk(`) are uniquely determined. Fork = 0, Eqs. (15)–(18) reduce to Roberts’ approximation for theutilization probabilities, cf. see above. It can efficiently be computed by determining any nontrivial solutionq

(h)0 (`) of the homogeneous recursion, e.g. with initial valueq

(h)0 (0) = 1, and then normalizing it via (18).

For obtainingqk(`) for k > 0, first one computes any particular solutionq(i)k (`) of the inhomogeneous

recursion (15)–(17) and any nontrivial solutionq(h)k (`) of the homogeneous recursion (15) and (16), where

sk(`) := 0,` = 0, . . . , C. As initial values one can choose, e.g.q(i)k (0) = 0 andq(h)

k (0) = 1. By inductionon` it follows that the sequencesq(i)

k (`)+ t q(h)k (`), ` = 0, . . . , C, parameterized byt ∈ R are the general

solution of the inhomogeneous recursion (15)–(17). Applying (18) one can determinetk ∈ R such that

qk(`) = q(i)k (`) + tkq

(h)k (`), ` = 0, 1, . . . , C, (19)

holds, i.e.

tk = %1∑C

`=C1−b1+1qk−1(`) −∑C`=0q

(i)k (`)∑C

`=0q(h)k (`)

, k = 1, 2, . . . . (20)

Finally, in view ofqk(`) ≥ 0, ` = 0, 1, . . . , C, we replaceqk(`) by max(qk(`), 0). Eq. (8) suggests thefollowing approximationfk for thekth factorial overflow moment of the first overflow stream:

fk := %1

C∑`=C1−b1+1

qk−1(`), k = 1, 2, . . . . (21)


The resulting algorithm forfk is of complexity O(Cmk) and, hence, very efficient compared to thenumerical computation offk. For k = 1 we obtain Roberts’ approximationpb,1 for the individualblocking probability:

pb,1 :=C∑

`=C1−b1+1

q0(`) = f1

%1. (22)

In case of equal bandwidths and equal mean holding times approximation (21) is exact.

4. Numerical examples and discussion

We consider a link with trunk reservation of capacityC = 100 offered tom = 3 traffic streams. In thefollowing examples we have%1b1 = 32,%2b2 = 48,%3b3 = 8 and, thus, a load of 88%. In Tables 1–4the trunk reservation parameters areC1 = 100,C2 = 96 andC3 = 94, providing a protection of thefirst stream. In Table 1 we consider the case of equal bandwidths and equal mean holding times, thus theapproximation for the factorial overflow moments is exact. In Table 2 only the mean holding times andcorrespondingly the arrival rates are changed, having no influence on the approximation for the blockingprobabilities. However, as the blocking probability for the protected first stream changes significantly, theapproximation for the factorial overflow moments of the first stream becomes worse. The approximationis better for the other streams, which have a larger blocking probability. This may be caused by the fact

Table 1Exact approximation in case of equal bandwidths and equal mean holding times

i λi µi bi Ci pb,i pb,i f1 f1 f2 f2 f3 f3

1 16.000000 1.000000 2 100 0.005943 0.005943 0.095092 0.095092 0.055604 0.055604 0.056150 0.0561502 24.000000 1.000000 2 96 0.081394 0.081394 1.953464 1.953464 7.917987 7.917987 43.452459 43.4524593 4.000000 1.000000 2 94 0.149649 0.149649 0.598594 0.598594 0.647483 0.647483 0.896254 0.896254

Table 2Approximation in case of equal bandwidths


1 8.000000 0.500000 2 100 0.002194 0.005943 0.035105 0.095092 0.010227 0.035669 0.005485 0.0224552 24.000000 1.000000 2 96 0.074262 0.081394 1.782289 1.953464 7.117383 8.232954 39.163192 47.1464433 8.000000 2.000000 2 94 0.153110 0.149649 0.612440 0.598594 0.880268 0.853101 1.620706 1.560586

Table 3Approximation in case of equal mean holding times


1 8.000000 1.000000 4 100 0.034824 0.038360 0.278593 0.306876 0.200301 0.208859 0.210952 0.2015562 24.000000 1.000000 2 96 0.095572 0.092728 2.293730 2.225468 10.193891 8.929272 60.140051 47.0219853 8.000000 1.000000 1 94 0.127782 0.124060 1.022253 0.992481 1.907904 1.679278 4.609830 3.638759


Table 4Approximation in the general case


1 4.000000 0.500000 4 100 0.022577 0.038360 0.180616 0.306876 0.081722 0.151186 0.056188 0.0984032 24.000000 1.000000 2 96 0.089264 0.092728 2.142336 2.225468 10.130069 8.633632 64.917762 43.7228793 16.000000 2.000000 1 94 0.130399 0.124060 1.043189 0.992481 2.750878 2.069166 9.427395 5.621440

Table 5Approximation in case of a link without trunk reservation

i λiµi biCi pb,i pb,i f1f1 f2f2 f3f3

1 4.000000 0.500000 4 100 0.097735 0.097735 0.781884 0.781884 1.092457 0.887371 2.000295 1.2556122 24.000000 1.000000 2 100 0.046515 0.046515 1.116367 1.116367 3.341622 2.469234 14.097474 7.4176583 16.000000 2.000000 1 100 0.022654 0.022654 0.181231 0.181231 0.154992 0.093722 0.200647 0.068983

that (11) holds in the mean, cf. (9). In Table 3 only the bandwidths and correspondingly the arrival ratesare changed compared to Table 1. The approximation depends on the bandwidths, and the approximationfor the factorial overflow moments of the first stream is much better than in case of changing the meanholding times, cf. Table 2. In Table 4 the mean holding times and the bandwidths are changed compared toTable 1. The approximation for the factorial overflow moments is also still better than in case of Table 2. InTable 5 we use the parameters of Table 4 but without trunk reservation. Since there is no trunk reservation,the approximation for the blocking probabilities and thus for the first overflow moments is exact, but theapproximation for higher overflow moments is obviously not exact.

From the given numerical examples as well as from numerical and simulation experiments not reportedhere we found that the quality of the proposed approximation for the first three individual overflowmoments is comparable to the quality of Roberts’ approximation for the individual blocking probabilities.In particular, the approximation becomes worse if the mean holding times greatly differ from each other.This situation may occur, since in some applications wideband calls may have both significantly longerholding times and significantly smaller arrival rates than narrowband calls. In a system integrating voiceand video conference calls, one expects that the video calls arrive relatively infrequently but have relativelylong durations. In these applications there are typically many narrowband arrivals and departures beforethe next wideband arrival or departure. Based on this disparity in time scales one can use the theory ofnearly completely decomposable Markov chains [28] for deriving appropriate approximations in this case,cf. [5, pp. 37–39] or [29] for this technique in case of a multiplexing system. However, as the stochasticdependence decreases in case of links with a larger number of traffic streams, the proposed approximationfor the factorial overflow moments is probably better in case of links with more thanm = 3 traffic streams.Further we conjecture that the proposed approximation for the second (and higher) overflow momentscan still be improved by using an improved approximation for the equilibrium utilization probabilities asinitialization of the recursion (15)–(18).

Since the approximation for the variance can be computed very fast, one can extend well-knownalgorithms, cf. [22,23,30], for computing performance measures of a circuit switched communicationnetwork based on a two moment characterization of traffic streams to networks containing links with


TRS. If a TRS link is a terminal link, then our algorithm can be integrated into the algorithms, where thedifferent parcel streams can be streamwise apportioned, e.g. by Wallstrøm’s formula, cf. [18, p. 26]. In caseof non-Poisson arrival streams further research is necessary for obtaining corresponding approximations.However, this seems to be a difficult problem. The ideas used in [23] for links without TRS cannotbe extended directly to TRS links as simulation studies have shown. The reason is that the TRS levelsinfluence extremely sensitive the overflow streams. However, a first-order approximation can be obtainedby replacing the streams arriving at a TRS link by Poisson streams. The advantage then is that one canuse the two moment technique at least for the links without TRS and for the terminal links in a network.The sketched ideas have to be outlined in detail in further research.

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[21] J.F.E. Wallin, B. Sanders, The calculation of overflow moments in loss systems with selective trunk reservation, Perform.Eval. 15 (1992) 195–202.

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factors and capacity requirements, IEEE Trans. Comm. COM-31 (1983) 1209–1211.[27] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer, New York, 1993.[28] P.J. Courtois, Decomposability — Queueing and Computer System Applications, Academic Press, London, 1997.[29] M.I. Reiman, J.A. Schmitt, Performance models of multirate traffic in various network implementations, in: J. Labetoulle,

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[30] CCITT, Recommendation E.524, The International Telegraph and Telephone Consultative Committee (CCITT), 1992,pp. 1–11.

Andreas Brandt was born in Berlin in 1955. He received his Ph.D. degree in mathematics from theHumboldt-University of Berlin in 1983. From 1983 to 1989 he was research assistant and from 1990to 1992 head of the group Applied Probability at the Department of Mathematics. Since 1992 he hasbeen Professor for Operations Research at the Humboldt-University of Berlin. His main research interestsare in stochastic processes, in particular point processes, and their applications. Over the last years hehas worked closely with several industrial partners. The projects address the performance analysis oftelecommunication and computer systems as well as reliability analysis of technical systems.

Manfred Brandt received his Ph.D. degree in mathematics from the Humboldt-University of Berlin in1973. Since 1989 he has been a Privatdozent at the Humboldt-University of Berlin. His main researchinterests are in pure complex analysis and since 1980s, also in the performance analysis of computer andteletraffic systems as well as queuing theory. He works closely with industrial partners and has implementedvarious algorithms. Since 1994 he has been a member of the scientific staff at the Konrad-Zuse-Zentrumfor Information Technology in Berlin.

Documents

Approximation for overflow moments of a multiservice link with trunk reservation