Convexity of Functions Which are Generalizations of the Erlang Loss Function and the ErlangDelay Function: Problem 90-8Author(s): A. A. Jagers and E. A. van DoornSource: SIAM Review, Vol. 32, No. 2 (Jun., 1990), pp. 301-302Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030528 .

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PROBLEMS AND SOLUTIONS 301

(i) If ABC is equilateral and the ratios r(DBC)/M(DBC), r(ECA)/M(ECA), r(FAB)/M(FAB) (which never can exceed 1/(2 13 - 0.28867) are all strictly greater than

c = (2 - )/2 - 0.1339746, prove that there can be no conic passing through the points A, B, C, D, E, F.

(ii) Find the smallest constant c' ' c, such that if ABC is no longer equi- lateral, but all four ratios r(ABC)/M(ABC), r(DBC)/M(DBC), r(ECA)/M(ECA), r(FAB)/M(FAB) are strictly greater than c', then again there can be no conic through the points A, B, C, D, E, F.

This problem arose in the study of techniques for interpolating between data scattered in the plane.

Convexity of Functions Which are Generalizations of the Erlang Loss Function and the Erlang Delay Function

Problem 90-8, by A. A. JAGERS AND E. A. VAN DOORN (Universiteit Twente, Enschede, the Netherlands). For real X, a, x with X, a > 0, let

f(x, a) =1() e-at tx-((l + t)x-x+l dt,

a confluent hypergeometric or Kummer function. Prove that, for X ' 1, log f(x, a) and f(x, a)-' are convex functions of x, for x ' 0. Or, equivalently, prove that for x? l,x0

(f (x, a))2 :f(x, a)fC"(x, a) _ 2(f (x, a))2, where a prime denotes differentiation with respect to x.

For X = 1, 2 two celebrated functions from teletraffic theory are obtained, viz., for x a nonnegative integer

af1 (x, a) =a ( a) >j= B(x, a)-',

the reciprocal of the Erlang loss function, while for x a nonnegative integer and x > a

af2(x, a) = I + ax E a

= Qxx) a)-'i!

the reciprocal of the Erlang delay function (cf. [1]). For X = 1 the problem has been solved in [3] (the error on line 4 of p. 45 may be

corrected by replacing a2 by (d/dt)2). For X = 2 only a weaker result is available in the literature. It has been shown in [2], and in a more general context in [4], that C(x, a)/(x - a) is a convex function of x for x > a and x a nonnegative integer. Clearly, the latter result is implied by the convexity of f2(x, a)' since C(x, a) is decreasing in x, x > a.

REFERENCES

[1] R. B. COOPER, Introduction to Queueing Theory, Second edition, Edward Arnold, London, 1981. [2] M. E. DYER AND L. G. PROLL, On the validity of marginal analysis for allocating servers in MIMIc

queues, Management Sci., 23 (1977), pp. 10 19-1022.

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302 PROBLEMS AND SOLUTIONS

[3] A. A. JAGERS AND E. A. VAN DOORN, On the continued Erlang loss function, Oper. Res. Lett., 5 (1986), pp. 43-46.

[4] R. R. WEBER, On the marginal benefit of adding servers to GIGI/m queues, Management Sci., 26 (1980), pp. 946-950.

An Optimization Problem

Problem 90-9*, by E. PINCHEON and C. C. ROUSSEAU (Memphis State University). Prove or disprove that for positive integers a, b, c satisfying a $ b, (a - b)2 < a + b,

and

a 2+b2-(a+b) c2(a +b) -(a -b)2

the discrete function

f a) b ) b)( a )X ,l ,C

is maximum at x = Lc/2J. This problem occurs in connection with the following question in graph theory. If G is a graph of order c, what is the largest possible number of induced subgraphs of G which are isomorphic to the complete bipartite graph K(a, b)?

Minimum Value of an Integral

Problem 90-10, by K. S. MURRAY (Brooklyn, NY). Determine the minimum value of

I= fn F'(t) + tm dt

where F'(t) ' 0, F(O) = a, F( 1) = b, n is a constant greater than 1, and m is a constant greater than or equal to zero.

Editor's note. To liven up the problem section, we invite readers to submit "Quickies" (preferably of an applied nature). Mathematical Quickies were initiated by C. W. Trigg in 1950 when he was editor of the problem section in Mathematics Magazine. These are problems that can be solved laboriously, but with proper insight and knowledge can be disposed of quickly. The above problem is a Quickie. In subsequent issues, these problems will not be identified as such except in their solutions appearing at the end of the same problem section.

SOLUTIONS

Problem of the Nile

Problem 89-6, by LEOPOLD FLATTO and LARRY SHEPP (AT&T Bell Laboratories). Let A be a two-dimensional set of finite Lebesgue measure. Then the probability

of A under the bivariate Gaussian distribution with covariance p is given by

I(P)=2 2fexp{2P2<l1 < dxdy 1 .

If I(p) is constant for -1 < p < 1, prove that X(A) = 0, where X denotes Lebesgue measure.

Remarks. (1) The finiteness of X(A) is crucial. For example, if A is a half-plane whose boundary contains the origin, then I(p) = 2 for -1 < p < 1.

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