Evaluation of the Erlang-B formula

Ondřej HudousekDepartment of Telecommunication engineering

Czech Technical University in Prague, Faculty of Electrical EngineeringTechnická 2

166 27 Praha 6 - DejviceE-mail: hudouso@feld.cvut.cz

Abstract

This article sums up the methods of evaluation of

the Erlang-B formula. In addition to known appro-

aches, some improvements of the evaluation leading

to its acceleration are introduced.

1 Introduction

Erlang-B is probably the most important formulaused for dimensioning of the loss systems classifiedM/M/N/0 according to Kendall’s classification. Itsaim is to determine the probability of loss basedon the offered traffic and the number of circuits. Itis often necessary to evaluate it with a prescribedaccuracy. As it is used repeatedly in some appli-cations, the speed of the evaluation might be signi-ficant.

The basic form of the Erlang-B formula for aninteger number of circuits is:

E1,N (A) = BN =AN

N !∑N

i=0Ai

i!

, (1)

where B is loss probability,A offered traffic,N number of circuits.

A generalized version of the formula is available fora non-integer number of circuits:

E1,x(A) =Axe−A

∫

∞

Atxe−tdt

. (2)

2 Evaluation of E1,N(A) for in-teger number of circuits

Direct evaluation of E1,N (A) according to thedefinition would be time consuming. Therefore, arecurrent form of the formula is used widely:

E1,N+1(A) = BN+1 =A ·BN

N + 1 +A ·BN

, B0 = 1.

(3)

Concerning a real implementation it is better toevaluate the reciprocal value of E1,N (A):

IN (A) =1

E1,N (A)= 1 +

N

AIN−1, I0 = 1. (4)

Due to constant offered traffic A during the eva-luation it is possible to multiply by the recipro-cal value 1/A instead of dividing by the value A.This accelerates the evaluation considerably as it isshown in the table1. It is also possible to substitutethe incrementation of a temporary variable with1/A for multiplication N(1/A), which also resultsin faster evaluation.

n 30 60 90 120 480 1890td[µs] 2.1 4.1 6.2 8.2 32.6 128.2to[µs] 0.8 1.4 2.0 2.7 10.4 40.9

Table 1: Time of evaluation of the Erlang-B2

1Duration of the evaluation was measured using RDTSCinstruction (see [6]). Due to a wide variety of availableprocessors and other impacts,the character of the above-mentioned values is only informative. The measuring wasperformed on Intel Pentium III running at 800 MHz, usingMS Windows 2000 and time critical priority of the process.2td – time of evaluation for direct implementation of the

recurrent formula, to – optimized version, N stays for num-ber of circuits. Loss probability is B = 0.01.

Another approach to fastening the evaluation isvia the approximation according to [1]. It can beshown that:

I0 = 1, (5)

I1 = 1 +1

A, (6)

I2 = 1 +2

A

(

1 +1

A

)

= 1 +2

A+1 · 2

A2, (7)

· · ·

IN = 1 +N

A+

N(N − 1)

A2+ · · ·+

N !

AN(8)

= 1 +

N∑

i=1

1

Ai

N !

(N − i)!. (9)

It is not necessary for large A and N to count in allterms of the sum in the equation (9). The difficultyof this approach lies in finding out the appropri-ate number of terms to achieve a desired accuracy.It is possible to accelerate the algorithm partiallyeven by checking whether the value IN is not higherthan the predefined value 1/E1,N . This approach issuitable for very large values of the offered traffic.

3 Evaluation of E1,x(A) for

non-integer number of cir-

cuits

Instead of direct evaluation of E1,x(A) for x ∈ R,x ≥ 0 various approximations are often used. Themost popular is probably the following one.

3.1 Rapp’s approximation

Y.Rapp published in [3] approximation based onthe equality of E1,x(A) and its first derivative withthe approximating function and its first derivativein the points x = 0 and x = 1:

E1,x(A) ≈ C0 + C1x+ C2x2, (10)

C0 = 1, (11)

C1 = −A+ 2

(1 +A)2 +A, (12)

C2 =1

(1 +A)[(1 +A)2 + A]. (13)

The approximating function is directly used in theinterval x ∈ 〈0..1〉. For x > 1 a fractional part ofthe loss probability is determined. The final valueof the approximation is evaluated using equation(3) which is also valid for N ∈ R.

3.2 Szybicky’s approximation

Another approximation was developed by Ed-mund Szybicky [4]. It is intended to be used on in-terval x ∈ 〈0..2〉 and it equals to E1,x(A) in pointsx = 0, 1, 2.

E1,x(A) ≈(2− x) ·A+A2

x+ 2A+A2. (14)

For x > 2 it is again possible to use the approxi-mation in conjunction with the recurrent formula(3).

3.3 Expansion of incomplete gamma

function into continued fraction

G.Lévy-Soussan suggested in [2] using expan-sion of incomplete gamma function into continuedfraction.

3.3.1 Continued fraction

According to Lévy-Soussan the expansion intocontinued fraction is function f(x0, x1, . . . , xn) inthe following form:

f(x0, x1, . . .) = b0 +a1

b1 +a2

b2 +a3

. . .

, (15)

which is sometimes written as

f(x0, x1, . . .) = b0+a1b1+

a2b2+ . . .+

an

bn+ . . . ,

(16)where ai and bi are functions of the real or com-plex variable xi. Limited expansion into the conti-nued fraction with n terms is function

fn(x0, x1, . . . , xp) = b0 +a1b1+

a2b2+ . . .+

an

bn.

(17)It is possible to evaluate fn directly from the end ofthe continued fraction. Unfortunately, this method

is suitable only when the number of the expansionterms is already known. Otherwise a recurrent for-mula suggested by J.Wallis in 1655 could be used:

fn =pn

qn

, (18)

pn = bnpn−1 + anpn−2, (19)

qn = bnqn−1 + anqn−2,

p−1 = 1, p0 = b0, q−1 = 0, q0 = 1.

3.3.2 Expansion of incomplete gamma

function

Incomplete gamma function is defined by theequation

Γ(a, x) =

∫

∞

x

e−tta−1dt, (20)

where Re(a) > 0 and Re(x) > 0. It is possible toexpress E1,x(A) as

E1,x(A) =Axe−A

Γ(1 + x, A). (21)

Incomplete gamma function could be expanded intocontinued fraction:

Γ(1 + x, A) = (22)

= e−AAx( 11+−xA+11+1− x

A+21+ . . .

)

,

where

a2n = −x+ n− 1, b2n = A,

a2n+1 = n, b2n+1 = 1,

a1 = 1, b0 = 0.

Above mentioned expansion into continued fractionis converging for all A > 0 and x > 0 (both condi-tions are valid for the real systems). Nevertheless,the convergence for x > A is slow, and therefore, itis often time-consuming to obtain a result with anappropriate accuracy. In these cases, it is advisa-ble to evaluate E1,A−1+θ(A), where θ is fractionalpart of x. The final value of the loss probability isevaluated with the recurrent equation (3) or (4).To achieve a desired accuracy it is advisable toevaluate separately expansion for even and odd

number of terms of the expansion rather than directexpansion of Fn(A, x). According to Lévy-Soussane

E1,x(A) = limn→∞

∣

∣

∣

∣

1

Pn(A, x)∼

1

In(A, x)

∣

∣

∣

∣

. (23)

It is possible to derive

Pn(A, x) =A

A− x+

xA− x+ 2

+ (24)

+2(x− 1)

A− x+ 4+ · · ·+

(n− 1)(x− n+ 2)A− x+ 2n− 2

,

In(A, x) = 1 +x

A− x+ 1+

x− 1A− x+ 3

+ (25)

+2(x− 2)

A− x+ 5+ · · ·+

(n− 1)(x− n+ 1)A− x+ 2n− 1

.

3.3.3 Evaluation of In(A, x) and Pn(A, x)

In [2] Lévy-Sousanne is only concerned with theevaluation of continued fraction backwards fromthe term bn. He comments on the usage of theequation (19): „. . . stability is not guaranteed andthe result obtained is the ratio of two almost-equallarge numbers, which is not acceptable in numeri-cal analysisÿ. As it has been stated above, it is ne-cessary to repeat the evaluation from the n–th termfor increasing n until reaching the desired accuracy.Obviously, this is a rather tedious procedure whichcompensates the advantage of rapid convergence ofthe functions In a Pn. However, the algorithm (19)was improved into the form of the modified Lentz’smethod [5]. Instead of using pn and qn temporaryvariables un and vn are used

un =pn

pn−1

, vn =qn−1

qn

(26)

and the resulting function is

fn = fn−1unvn, (27)

where

f0 = b0, u0 = f0 = b0, v0 = 0. (28)

un =pn

pn−1

= bn + an

pn−2

pn−1

= bn +an

un−1

, (29)

vn =(

bn + an

an−2

an−1

)

−1

= (bn + anvn−1)−1. (30)

A problem may occur if the denominator in theevaluation of un or vn equals to zero. Thompsonand Barnett suggested a slight modification of thealgorithm to address this issue. If un−1 = 0 orbn+ anvn−1 = 0 during the evaluation, it is substi-tuted for a small constant, e.g. 10−30. Lentz’s algo-rithm could be symbolically written for the desiredaccuracy ε as follows:

1 tiny ← 10−30,2 f0 ← b0,3 if b0 = 0, then f0 ← tiny,4 u0 ← f0,5 v0 ← 0,6 ∆j ← ε,7 while |∆j − 1| < ε, do:8 vj ← bj + ajvj−1,9 if vj = 0, then vj ← tiny,10 uj ← bj + aj/uj−1,11 if uj = 0, then uj ← tiny,12 vj ← 1/vj ,13 ∆j ← ujvj ,14 fj ← fj−1∆j .

From the practical point of view, it is advisableto evaluate Pn(A, x) and In(A, x) simultaneously,as it is possible to exploit the similarity of the ex-pansion terms, (see (24),(25)) and thus acceleratethe algorithm.

3.4 Comparison of the evaluation

time

To discover the practical suitability of the algo-rithms, their speed has been examined. Table 2shows the results for both Rapp’s approximation(tr and tropt) and the expansion of incomplete ga-mma function into continued fraction (tc and tcopt,accuracy 1.10−5, B=0.01) both for direct and opti-mized implementation:

n 30 60 90 120 480 1890tr[µs] 4.6 8.8 13.0 17.2 67.9 266.3

tropt[µs] 0.6 0.9 1.2 1.5 5.2 19.5tc[µs] 5.4 7.0 7.9 8.5 12.4 16.3

tcopt[µs] 3.3 4.1 5.2 5.8 11.6 28.8Table 2: Comparison of the times spent on

evaluation of the Erlang-B formula using variousimplementations.

4 Conclusion

The aim of this article is to sum up the me-thods of evaluation of the Erlang-B formula. In ad-dition to known approaches, usage of Lentz’s algori-thm for evaluation of the expansion into continuedfraction was suggested. The usage of the expansionof incomplete-gamma function brings, in contrastto the Rapp approximation, feasibility to meet thepredefined accuracy. This advantage is compensa-ted by higher complexity of the algorithm, whichresults in slower evaluation for small numbers ofcircuits, as it was proved by testing on real imple-mentations. On the testing system , the optimizedversion of Rapp’s approximation was faster thanevaluation of the expansion of incomplete gammafunction up to approx. 1500 circuits (B=0.01). The-refore, Rapp’s approximation is probably the mostsuitable for common cases, as far as the evaluationtime. Apart from that, some improvements leadingto fastening the evaluation of the recurrent formula(3) have been suggested.

Reference

[1] ITC; ITU-D SG2. Teletraffic en-

gineering handbook. Available at<http://www.tele.dtu.dk/teletraffic>.

[2] Lévy-Soussan, Guy. Numerical Evaluation ofthe Erlang Function through a Continued-Fraction Algorithm. Electrical Communi-cation, 1968, vol.43, no.2, p.163–168.

[3] Rapp, Yngve. Planning of Junction Network ina Multi–exchange Area. II Extensions of thePrinciples and Applications. Ericsson Tech-nics, 1965, no.2, p.187–240.

[4] Szybicky, Edmund.Some Numerical MethodsUsed for Telephone Traffic Theory Applicati-ons. Ericsson Technics, 1964, no.2, p.203-229.

[5] Numerical Recipes in C: The Art of ScientificComputing, Cambridge University Press, 1992.Available at <http://www.nr.com>.

[6] Using the RDTSC Instrucion for Per-

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