0,4

When s is a positive integer and 0<rho<1 then

C(s,rho):=(s*rho)^s/G(s,rho)/s is the well known Erlang delay (or the

Erlang's C) formula. This measure is a basic formula of queueing theory. The

applications of this formula are in diverse systems where queueing phenomena

arise, including telecommunications, production and service systems. The

formula gives the steady-state probability of delay in the M/M/s queueing

system. The number of servers is denoted by s and the traffic intensity is

denoted by rho, 0<rho<1, where rho=(arrival rate)/(service rate)/s.

Cooper, R. B. 1981, Introduction to Queueing Theory. Second ed., North Holland, New York.

Harel, A. 1988. Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas. Management Science, Vol. 34, 959-972.

Harel, A. and P. Zipkin. 1987a. Strong Convexity Results for Queueing Systems. Operations Research, Vol. 35, No. 3, 405-418.

Harel, A. and P. Zipkin. 1987b. The Convexity of a General Performance Measure for the Multi-Server Queues. Journal of Applied Probability, Vol. 24, 725-736.

Jagers, A. A. and E. A. van Doorn, 1991. Convexity of functions which are generalizations of the Erlang loss function and the Erlang delay function. SIAM Review. Vol. 33 (2), 281-282.

Lee, H. L. and M. A. Cohen. 1983. A Note on the Convexity of Performance Measures of M/M/c Queueing Systems. Journal of Applied Probability, Vol. 20, 920-923.

Medhi, J. 2003. Stochastic Models in Queueing Theory. Second ed., Academic Press, New York.

Smith, D.R. and W. O. Whitt. 1981. Resource Sharing for Efficiency in Traffic Systems. Bell System Technical Journal, Vol. 60, No. 1, 39-55.

Erlang, A. K. Erlang

An equivalent expression for G(s,rho) that is often used is: G(s,rho)=Sum(s^i*rho^i/i!,i=0..s-1)*(1-rho)*(s-1)!+rho^s*s^(s-1);

For s>0 and rho>0 one can use the expression: G(s,rho)=(exp(s*rho)*s*rho*(1-rho)*(s-1)*GAMMA(s-1,s*rho)+rho^s*s^s)/s/rho;

For s>0 and rho>0 one can also use the integral representation G(s,rho)=(s*rho)^s/s*Int(rho*s*exp(-rho*s*t)*t*(1+t)^(s-1), t=0..infinity);

G(5,rho)=24+96*rho+180*rho^2+200*rho^3+125*rho^4. The coefficients (24, 96, 180, 200, 125) give the 5th line of the triangle.

Triangle begins:

1;

1, 1;

2, 4, 3;

6, 18, 24, 16;

24, 96, 180, 200, 125;

120, 600, 1440, 2160, 2160, 1296;

720, 4320, 12600, 23520, 30870, 28812, 16807;

5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144;

40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969;

Sequence in context: A046793 A101278 A091274 this_sequence A054589 A051851 A011171

Adjacent sequences: A122522 A122523 A122524 this_sequence A122526 A122527 A122528

nonn,tabl

Arie Harel (Arie_Harel(AT)baruch.cuny.edu), Sep 14 2006