Explicit calculation of overflow variances in loss systems with selective trunk reservation

Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71 www.elsevier.de/aeue

Explicit calculation of overflow variances in loss systems with selective trunk reservation Rudolf G. Schehrer Communication Networks Institute, University of Dortmund, D44221 Dortmund, Germany Received 25 August 2007; accepted 28 January 2008

Abstract In this paper, systems with selective trunk reservation are investigated. A (primary) group with two offered traffic streams is considered. For the high-priority traffic the trunk group acts as a group of full accessibility. Calls of the low-priority traffic can only be switched if more than a number of nres trunks (nres trunks are reserved for the high-priority traffic) are idle. In the paper, explicit formulae for the variance of the low-priority overflow traffic and of the total overflow traffic are presented (formulae for the high-priority overflow traffic have been published earlier [Schehrer R. On the explicit calculation of overflow moments of the high-priority traffic in loss systems with selective trunk reservation. In: 19th international teletraffic congress, Beijing: Beijing University of Posts and Telecommunications Press; 2005. p. 1029–38]). The calculations are carried out in terms of factorial moments. 䉷 2008 Published by Elsevier GmbH Keywords: Trunk reservation; Overflow traffic; Variance

1. Introduction Since several years, systems with selective trunk reservation (STR) have found a wide spread interest, e.g., in mobile cellular communication systems. Such systems have been investigated by Wallin and Sanders [1]. Further literature can be found in [1]. In the system considered here, a high-priority traffic and a low-priority traffic are offered to a common trunk group. For the high-priority traffic, the trunk group has full accessibility, i.e., calls can be switched if there is at least one idle trunk. Calls of the low-priority traffic can, however, only be switched if there are more than nres idle trunks, i.e., nres > 0 can be considered as the number of reserved trunks, by which the high-priority traffic is protected against the low-priority E-mail addresses: [email protected], [email protected]. 1434-8411/$ - see front matter 䉷 2008 Published by Elsevier GmbH doi:10.1016/j.aeue.2008.02.001

traffic. The traffic streams overflowing behind the common trunk group may be offered to further trunk groups. For the loss calculations in such secondary trunk groups it is convenient to apply, e.g., two moment methods (i.e., variance methods) [2], taking into account the variance as well as the mean of overflow traffic streams. In [1], a recursive method for the calculation of the higher order moments of the overflowing traffic streams (and explicit formulae for the special case of only nres = 1 reserved trunk) is described. It is, however, well known that recursive calculations may lead to numerical instabilities. Therefore, explicit solutions are of particular importance for practical applications. In [3], explicit formulae have been described for the higher order moments of the high-priority overflow traffic. For the low-priority overflow traffic and for the total overflow traffic streams, the state equations are somewhat more complicated. Therefore, in the present paper explicit formulae are presented only for the mean value and the

66

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

variance, i.e., the first-order moment and the second-order moment (as the most important one of the higher order moments) for the low-priority overflow traffic and for the total overflow traffic. The explicit calculation of further higher order moments is somewhat more complicated and therefore beyond the scope of the present paper. This paper can be considered as a continuation of the paper [3] which has been presented at ITC 19. In Section 2 the variance of the low-priority overflow traffic is derived. Section 3 deals with the variance of the total overflow traffic. Numerical examples are also presented. A conclusion follows in Section 4.

2. Variance of the low-priority overflow traffic

Let the probability that x1 trunks are busy in the primary group be denoted as p1 (x1 ), and the probability that x2 trunks are busy in the secondary group by p2 (x2 ). Let the probability of a state with x1 busy trunks in the primary group and x2 busy trunks in the infinite secondary group be denoted by p(x1 , x2 ). Then the following relations hold true: p1 (x1 ) =

∞ 

p2 (x2 ) =

n 

In the system configuration considered here, a highpriority traffic of mean AH and a low-priority traffic of mean AL are offered to a common trunk group consisting of n trunks. Both offered traffic streams are considered to be of Poisson type, i.e. the interarrival times of arriving calls are negative exponentially distributed with the mean values H and L of the arrival rates, respectively, and the holding times have also a negative exponential distribution with the mean h (for both traffic streams). Calls of the low-priority traffic which cannot be switched in the trunk group are overflowing to a secondary group with an infinite number of trunks, n2 → ∞. The mean of this overflow traffic is denoted as RL (the other overflow traffic RH consisting of blocked calls of the high-priority traffic is not considered in this section). The number of busy trunks in the common trunk group is denoted by x1 (comprising both high- and low-priority calls) and the number of busy trunks in the infinite secondary groups by x2 (comprising only low-priority calls in this case). The mean values AH and AL of the offered traffic streams are

(3a)

p(x1 , x2 ).

(3b)

x1 =0

It is convenient for a suitable notation to define p(x1 , x2 ) = 0

2.1. System configuration

p(x1 , x2 ),

x2 =0

for x1 < 0 or x2 < 0 or x1 > n.

(3c)

With these notations the following equations of state are obtained [1]: Domain I (x1 < m): (x1 + x2 + A)p(x1 , x2 ) = Ap(x1 − 1, x2 ) + (x1 + 1)p(x1 + 1, x2 ) + (x2 + 1)p(x1 , x2 + 1). (4a) Domain II (x1 = m): (x1 + x2 + A)p(x1 , x2 ) = Ap(x1 − 1, x2 ) + (x1 + 1)p(x1 + 1, x2 ) + (x2 + 1)p(x1 , x2 + 1) + AL p(x1 , x2 − 1).

(4b)

Domain III (m < x1 < n): (x1 + x2 + A)p(x1 , x2 ) = AH p(x1 − 1, x2 ) + (x1 + 1)p(x1 + 1, x2 ) + (x2 + 1)p(x1 , x2 + 1) + AL p(x1 , x2 − 1).

(4c)

Domain IV (x1 = n): (x1 + x2 + AL )p(x1 , x2 ) = AH p(x1 − 1, x2 ) + AL p(x1 , x2 − 1) + (x2 + 1)p(x1 , x2 + 1).

(4d)

AH = H h,

(1a)

AL = L h.

(1b)

In addition to the set of Eqs. (4a–d), the normalizing condition holds true  p(x1 , x2 ) = 1. (4e)

For the total offered traffic A and for the total overflow traffic R holds

Furthermore, the following global (or, generalized) state equation holds true:

A = AH + AL , R = RH + RL .

(1c) (1d)

Let the number of trunks which are accessible for the lowpriority traffic be denoted as m, where m = n − nres .

(2)

AL

n 

p(x1 , x2 ) = (x2 + 1) p2 (x2 + 1).

(5)

x1 =m

This global state equation can also be obtained by summing up Eqs. (4a–d) for all values of x1 and for the values x2 = 0, . . . ,  and then replacing  by x2 .

2.2. Equations of state

2.3. Calculation of the moments

In the general case, the states of the system can be subdivided into four domains. Domain I for x1 < m, domain II for x1 =m, domain III for m < x1 < n and domain IV for x1 =n.

2.3.1. Equations for the moments The ordinary moments mr (x2 ) of order r and the factorial moments Mr (x2 ) of the traffic overflowing to the secondary

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

group are defined as mr (x2 ) =

∞ 

x2r p2 (x2 ),

(6)

Mr (x2 ) =



∞  x2 =0

x2 r

 (7)

r!p2 (x2 ).

For r = 0 and 1, respectively, holds (8)

m1 (x2 ) = M1 (x2 ) = RL .

(9)

The central moments r (x2 ) and the variance are defined as r (x2 ) =

[x2 − m1 (x2 )] p2 (x2 ), r

∞ 

[x2 − m1 (x2 )]2 p2 (x2 ).

(10)

  ∞  Ax1 −v r −1+v . Sr,x1 (A) = v (x1 − v)!

(11)

The S-polynomials according to Eq. (19) fulfil the conditions [5] Sr,x = Sr−1,x + Sr,x−1 , Sr+1,x =

x2 =0

Mr (x2 |x1 ).

(13)

x1 =0

  Multiplying Eq. (5) by the factor xr2 r! and summing up for all values x2 0 and regarding Eqs. (12), (13) and (14) leads (after rearrangement of terms) to relations (15) and (16) [4]     x x2 + 1 (r + 1)!, (14) (x2 + 1) 2 r! = r +1 r Mr+1 (x2 ) = AL

n 

Mr (x2 |x1 ),

r 0,

(15)

Mr−1 (x2 |x1 ),

r 1.

(16)

x1 =m

Mr (x2 ) = AL

n  x1 =m

According to Eq. (16), the moment M2 (x2 ) can be expressed as a function of the conditional factorial moments M1 (x2 |x1 ). Next, the equations for the conditional factorial moments have therefore to be considered. 2.3.2. The conditional moments   Domain I : Now Eq. (4a) is multiplied by the factor xr2 r! and summed up for all values of x2 0 as shown above. After rearrangement of terms the following equation is obtained: (x1 + A + r)Mr (x2 |x1 ) = (x1 + 1)Mr (x2 |x1 + 1) + AMr (x2 |x1 − 1), 0 x1 < m,

x 

Sr, ,

Sr,0 = 1, S0,x =

For the moments known as conditional factorial moments of order r holds  ∞   x2 Mr (x2 |x1 ) = r!p(x1 , x2 ), (12) r n 

(19)

(20) (21)

=0

x2 =0

Mr (x2 ) =

(18)

where

x2 =0

VL = 2 (x2 ) =

0 x1 m,

v=0

m0 (x2 ) = M0 (x2 ) = 1,

∞ 

where of course Mr (x2 | − 1) is equal to zero. According to Brockmeyer [5], this difference equation (17) has the solution Mr (x2 |x1 ) = Mr (x2 |0)Sr,x1 (A),

x2 =0

67

(17)

Ax . x!

(22a) (22b)

(The argument A of the function Sr,x is omitted here for the sake of simplicity.) These S-polynomials can be evaluated easily by means of successive additions according to Eq. (20), starting with the S-polynomials for r = 0 and m = 0 according to the Eqs. (22a) and (22b), respectively [5]. Although this paper aims at avoiding recursive calculations, such a recursive calculation of the S-polynomials is acceptable because no subtractions are involved (and thus there is no danger of numerical instabilities). Eq. (18) yields for x1 = m − 1 and x1 = m, respectively Mr (x2 |m − 1) = Mr (x2 |0)Sr,m−1 (A),

(23a)

Mr (x2 |m) = Mr (x2 |0), Sr,m (A).

(23b)

(For x1 =m−1, Eq. (17) yields an equation for the calculation of the moment Mr (x2 |m) from the values Mr (x2 |m − 1) and Mr (x2 |m − 2). Therefore Eqs. (18) and (23b) are also valid for x1 = m). Domain II : Arguing along the same lines as in case of domain I, Eq. (4b) for x1 = m leads to the equation (m + AH + r)Mr (x2 |m) = (m + 1)Mr (x2 |m + 1) + AMr (x2 |m − 1) + rAL Mr−1 (x2 |m)

(24)

and with Eq. (23a,b) to the equation Mr (x2 |m + 1) 1 = [(m+AH +r)Sr,m (A) − ASr,m−1 (A)]Mr (x2 |0) m+1 1 − (25) rA2 Mr−1 (x2 |0)Sr−1,m (A). m+1 Domain III : Analogously, Eq. (4c) leads to the following equation if the conditional factorial moments of domain III

68

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

  AH A S1,m (A) − S1,m−1 (A), D2 = 1 + m+1 m+1

are denoted by Mr∗ (x2 |x1 ) (x1 + AH + r)Mr∗ (x2 |x1 ) = (x1 + 1)Mr∗ (x2 |x1 + 1) + AH Mr∗ (x2 |x1 ∗ (x2 |x1 ), m < x1 < n, + rAL Mr+1 Mr∗ (x2 |x1 ) = Mr (x2 |x1 ),

− 1)

D3 = (26a)

m x1 n.

(26b)

For x1 m, the values Mr∗ (x2 |x1 ) are identical to the values Mr (x2 |x1 ). For x1 < m, the values Mr∗ (x2 |x1 ) represent hypothetical values, which, however, can also be calculated according to Eq. (26a). Then it can be seen that, despite the fact that the conditional moments Mr (x2 | − 1) of domain I and M0∗ (x2 | − 1) are equal to zero, the hypothetical values Mr∗ (x2 | − 1) for r > 0 are not equal to zero. Therefore the S-polynomials according to Eq. (19) are a solution of Eq. (26a) for r = 0, but not for r > 0, i.e., not for the moments of first order. For the values M0∗ (x2 |x1 ), the following solution is obtained: M0∗ (x2 |x1 ) = M0∗ (x2 |0)S0,x1 (AH ). The condition that the values M0 (x2 |m) and identical leads to the equation

(27a) M0∗ (x2 |m)

M0∗ (x2 |x1 ) = dM0 (x2 |0)S0,x1 (AH ),

are

(27b)

M1 (x2 |m) = M0 (x2 |0) D1 C,

(31a)

M1 (x2 |m + 1) = M0 (x2 |0)(D2 C − D3 ).

(31b)

On the other hand, these values can be expressed with the aid of Eq. (28). M1∗ (x2 |m) = M1∗ (x2 |0)S1,m (AH ) − M1∗ (x2 | − 1)[S1,m (AH ) − 1] − dM0 (x2 |0)AL S1,m−1 (AH ),

(32a)

M1∗ (x2 |m + 1) = M1∗ (x2 |0)S1,m+1 (AH ) − M1∗ (x2 | − 1)[S1,m+1 (AH ) − 1] − dM0 (x2 |0)AL S1,m (AH ).

(32b)

With the aid of these two equations the unknown values M1∗ (x2 |0) and M1∗ (x2 |−1) can be calculated from the values M1∗ (x2 |m) and M1∗ (x2 |m + 1). With the abbreviations D4 = D5 =

(27c)

For domain III an explicit solution can be found for the values M1∗ (x2 |x1 ) by means of generating functions (in analogy to a method applied in [3]). In this way the following explicit solution is found: M1∗ (x2 |x1 ) = M1∗ (x2 |0)S1,x1 (AH ) − M1∗ (x2 | − 1)[S1,x1 (AH ) − 1] − dM0 (x2 |0)AL S1,x1 −1 (AH ).

C=

M1 (x2 |0) . M0 (x2 |0)

(29)

as a new value which is still unknown. Furthermore, for reasons of simplicity and in order to avoid lengthy terms in various equations, several abbreviations (D1 , D2 , etc.) are introduced in the sequel, D1 = S1,m (A),

(30a)

1 S0,m+1 (AH ) 1 S0,m+1 (AH )

[S1,m+1 (AH ) − 1],

(33a)

[S1,m (AH ) − 1],

(33b)

D6 =

1 AL [S1,m−1 (AH )(S1,m+1 (AH ) − 1) S0,m+1 (AH ) − S1,m (AH )(S1,m (AH ) − 1)], (33c)

D7 =

1 S1,m+1 (AH ), S0,m+1 (AH )

(33d)

D8 =

1 S1,m (AH ), S0,m+1 (AH )

(33e)

D9 =

1 AL [S1,m−1 (AH ) S1,m+1 (AH ) S0,m+1 (AH ) 2 − S1,m (AH )],

(28)

The derivation of this formula is omitted here. The validity of Eq. (28) can, however, easily be proved by means of induction. The values M1∗ (x2 |0) and M1∗ (x2 |−1) in Eq. (28) have to be determined such that the conditional moments for x1 = m and x1 = m + 1 in Eq. (28) are in accordance with values according to Eqs. (23b) and (25). The value M1 (x2 |0) is not yet known. It is convenient to introduce the abbreviation

(30c)

With these abbreviations, Eqs. (23b) and (25) for r = 1 can be written as follows:

with S0,m (A) . d= S0,m (AH )

AL S0,m (A). m+1

(30b)

(33f)

the solution of this set of two linear equations yields the results M1∗ (x2 |0) = M1 (x2 |m)D4 − M1 (x2 |m + 1)D5 + M0 (x2 |0)dD6 ,

(34a)

M1∗ (x2 | − 1) = M1 (x2 |m)D7

− M1 (x2 |m + 1)D8 + M0 (x2 |0)dD9 ,

(34b)

or, with Eqs. (31a, b) M1∗ (x2 |0) = [CD1 D4 − (CD2 − D3 )D5 + dD6 ]M0 (x2 |0),

(35a)

M1∗ (x2 | − 1) = [CD1 D7

− (CD2 − D3 )D8 + dD9 ]M0 (x2 |0).

(35b)

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

Finally, with the abbreviations D10 = D1 D4 − D2 D5 ,

(36a)

D11 = D3 D5 + dD6 ,

(36b)

D12 = D1 D7 − D2 D8 ,

(36c)

D13 = D3 D8 + dD9 ,

(36d)

the following results are obtained: M1∗ (x2 |0) = M0 (x2 |0)[CD10

(37b)

With these Eqs. (37a, b) in combination with Eq. (28) all conditional moments M1∗ (x2 |x1 ) of first order are now expressed as a function of the variable C and the value M0 (x2 |0) which can be considered as a normalizing constant. Domain IV : The equations referring to domain IV are not used for the calculation of the conditional moments and are therefore not considered here. 2.3.3. The moments Mr (x2 ) The moments Mr (x2 ) according to Eq. (13) can be written in the form: Mr (x2 ) =

Mr (x2 |x1 ) +

x1 =0

n 

Mr (x2 |x1 ),

(38)

(39)

x1 =m

For the factorial moment M0 (x2 ) of the order zero one obtains with Eqs. (27b) and (39)   n  S0,x1 (AH ) . (40) M0 (x2 )=M0 (x2 |0) S1,m−1 (A) + d x1 =m

With the abbreviations D14 = S1,m−1 (A), D15 = d

n 

S0,x1 (AH ),

(41) (42)

x1 =m

M0 (x2 ) = M0 (x2 |0)(D14 + D15 ) = 1

(43)

and thus 1 . D14 + D15

D15 . D14 + D15

(46)

On the other hand M1 (x2 ) can be determined according to Eq. (39). Inserting Eqs. (28) and (37a, b) in Eq. (39) with the abbreviations D16 = S2,m−1 (A), D17 =

n 

(47a)

S1,x1 (AH ),

(47b)

[S1,x1 (AH ) − 1],

(47c)

xi =m

D18 =

n  xi =m

D19 = dAL

n 

(47d)

S1,x1 −1 (AH )

yields

or

(44)

Now all values M0 (x2 |x1 ) are known according to Eqs. (18), (27b) and (44).



M1 (x2 ) = M0 (x2 |0)

CD16 + (CD10 + D11 )D17 −(CD12 + D13 )D18 − D19

(48a)

C(D16 + D10 D17 − D12 D18 ) . +D11 D17 − D13 D18 − D19 (48b)

With the abbreviations D20 = D16 + D10 D17 − D12 D18 ,

(49a)

D21 = D11 D17 − D13 D18 − D19 ,

(49b)

D22 = (D14 + D15 )M1 (x2 ),

(49c)

the following result for the variable C is obtained C=

the following equation is obtained:

M0 (x2 |0) =

or, with Eqs. (41) and (44)

M1 (x2 ) = M0 (x2 |0) Mr (x2 |x1 ).

(45)

S1,x1 (AH ),

xi =m



or, with Eqs. (18) and (21) n 

n 

x1 =m

x1 =m

Mr (x2 ) = Mr (x2 |0)Sr+1,m−1 (A) +

RL = M1 (x2 ) = AL dM0 (x2 |0)

(37a)

M1∗ (x2 | − 1) = M0 (x2 |0)[CD12 + D13 ].

m−1 

For the factorial moment M1 (x2 ), i.e., the mean of the traffic overflowing to the secondary group, follows from Eqs. (9), (16), (27b) and (42)

RL = M1 (x2 ) = AL

+ D11 ],

69

D22 − D21 . D20

(50)

Now all conditional moments M1 (x2 |x1 ) of first order can be calculated explicitly according to Eqs. (18) or (28) and (50), respectively. Then for r = 2 the second factorial moment M2 (x2 ) is obtained according to Eq. (16) as: M2 (x2 ) = AL

n  x1 =m

M1 (x2 |x1 ).

(51)

70

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

The variance of the traffic overflowing to the secondary group is now obtained as follows: if Eq. (7) for r = 2 and Eq. (11) are written out in full and combined, then the following explicit formula for the variance is foundas: VL = M2 (x2 ) + RL − RL2 .

(52)

3. Variance of the total overflow traffic In this section the variance V of the total overflow traffic R is considered which, e.g., can be applied in cases where the high-priority overflow traffic RH and the low-priority overflow traffic RL are offered to the same (finite) secondary group. In this section, the number x2 of busy trunks in the (infinite) secondary group comprises high-priority calls as well as low-priority calls. In the sequel, equation numbers with an asterisk are used. As compared to Section 2, the equations are partly different. The structure of the equations, however, is the same. In domain I, domain II and domain III the state equations are identical in both cases, whereas the equations in domain IV and the global state equations are different. Consequently, several further equations (concerning the moments) are also identical. In the sequel, for reasons of simplicity only those equations are listed which are not identical to the corresponding equations of Section 2.

AL

n−1 

(4d*)

p(x1 , x2 ) + Ap(n, x2 ) = (x2 + 1) p2 (x2 + 1), (5*)

x1 =m

m1 (x2 ) = M1 (x2 ) = R, V= 2 (x2 ) =

∞ 

(9*)

[x2 − R]2 p2 (x2 ),

M2 (x2 ) = AL

AL D15 + AH dS0,n (AH ) , D14 + D15

n−1 

M1 (x2 |x1 ) + AM 1 (x2 |n),

(11*)

(46*)

(51*)

x1 =m

V = M2 (x2 ) + R − R 2 .

A numerical example is contained in Section 3.

(x1 + x2 + A)p(x1 , x2 ) = Ap(x1 − 1, x2 ) + AH p(x1 , x2 − 1) + (x2 + 1)p(x1 , x2 + 1),

R = M1 (x2 ) =

(52*)

A complete set of the equations for the calculation of the variance of the total overflow traffic can be obtained by inserting the equations of Section 3 in Section 2 in such a way, that they replace the corresponding equations (with the corresponding equation numbers) in Section 2. Example: For a system with n=25 trunks, nres =4 reserved trunks and the offered traffic values AL = 12 Erlangs and AH = 12 Erlangs the following values for the low-priority overflow traffic and the total overflow traffic are obtained (for comparison, the corresponding values RH and VH of the high-priority overflow traffic according to [3] are also included) RL = 4.4979 Erlangs, VL = 9.7920, RH = 0.1505 Erlangs, VH = 0.2672, R = 4.6485 Erlangs, V = 10.7375.

4. Conclusion This paper deals with the variance of the high-priority overflow traffic and of the total overflow traffic in loss systems with selective trunk reservation. After establishing the equations of state, these equations are transformed into equations of the factorial moments. With the aid of S-polynomials according to [5] and generating functions these equations can be solved. Finally, explicit formulae for the second moments (and for the variance values) of the overflow traffic streams are obtained, which can, e.g., be used in applications of a two moments method [2].

References

x2 =0

Mr+1 (x2 ) = AL

n 

Mr (x2 |x1 )

x1 =m

+ AH Mr (x2 |n), Mr (x2 ) = AL

n 

r 0,

(15*)

r 1,

(16*)

Mr−1 (x2 |x1 )

x1 =m

+ AH Mr−1 (x2 |n), R = M1 (x2 ) = AL dM0 (x2 |0)

n 

S0,x1 (AH )

x1 =m

+ AH dM0 (x1 |0)S0,n (AH ),

(45*)

[1] Wallin JFE, Sanders B. The Calculation of overflow moments in loss systems with selective trunk reservation. Perform Eval 1992;15:195–202. [2] Wilkinson RI, (main paper) and Riordan J. (appendix), Theories for toll traffic engineering in the USA. (a) In: First international teletraffic congress, Copenhagen, 1955. Doc. 43; (b) Bell Syst Tech J (1956); 35: 421–514. [3] Schehrer R. On the explicit calculation of overflow moments of the high priority traffic in loss systems with selective trunk reservation. In: 19th international teletraffic congress. Beijing: Beijing University of Posts and Telecommunications Press; 2005. p. 1029–38. [4] Schehrer R. On higher order moments of overflow traffic behind groups of full access. In: Proceedings of the 8th international teletraffic congress, Melbourne, 1976. p. 422/1–8.

R.G. Schehrer / Int. J. Electron. Commun. (AEÜ) 63 (2009) 65 – 71

[5] Brockmeyer E. The simple overflow problem in the theory of telephone traffic. Teleteknik 1954;5:361–74 (in Danish) and monograph in English (1955).

Rudolf Schehrer was born in Eislingen, Germany, in 1939. He received the Dipl.-Ing. degree and the Ph.D. degree in communication engineering, both from the University of Stuttgart, Germany, in 1964 and 1969, respectively. From 1964 to 1970 he was a Research Assistant in the Institute for Switching and Data Technics at the University of

71

Stuttgart. Then he was with the Research Institute of the AEGTelefunken company in Ulm, Germany, from 1970 to 1978. Since 1979 he has been head of the Institute for Electronic Systems and Switching at the University of Dortmund, Dortmund, Germany, which has recently been renamed and is now called Communication Networks Institute. Since 2004 he is retired. His main interests are in the fields of teletraffic engineering and switching systems.