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A122525
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Triangle read by rows: G(s,rho)=(s-1)!/s*Sum((s-i)/i!*(s*rho)^i,i=0..(s-1));
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3
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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.
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REFERENCES
| 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.
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LINKS
| Erlang, A. K. Erlang
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FORMULA
| 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);
Multiplying the n-th row entries by n+1 results in triangle A066324 in row reversed form. - Peter Bala, Sep 30 2011
Row generating polynomials are given by 1/n*D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(x)/(1-x)*d/dx. - Peter Bala, Dec 27 2011
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EXAMPLE
| 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;
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CROSSREFS
| Cf. A066324.
Sequence in context: A182940 A101278 A091274 * A054589 A051851 A011171
Adjacent sequences: A122522 A122523 A122524 * A122526 A122527 A122528
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KEYWORD
| nonn,tabl
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AUTHOR
| Arie Harel (Arie_Harel(AT)baruch.cuny.edu), Sep 14 2006
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