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%I #52 Jan 25 2022 08:35:01
%S 1,1,1,2,4,3,6,18,24,16,24,96,180,200,125,120,600,1440,2160,2160,1296,
%T 720,4320,12600,23520,30870,28812,16807,5040,35280,120960,268800,
%U 430080,516096,458752,262144,40320,322560,1270080,3265920,6123600,8817984,9920232,8503056,4782969
%N Triangle read by rows: G(s, rho) = ((s-1)!/s)*Sum_{i=0..s-1} ((s-i)/i!)*(s*rho)^i.
%C 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.
%C With offset = 0, T(n,n-k) is the number of partial functions on {1,2,...,n} with exactly k recurrent elements for 0 <= k <= n. Row sums = (n+1)^n. - _Geoffrey Critzer_, Sep 08 2012
%D Cooper, R. B. 1981, Introduction to Queueing Theory. Second ed., North Holland, New York.
%D Harel, A. 1988. Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas. Management Science, Vol. 34, 959-972.
%D Harel, A. and P. Zipkin. 1987a. Strong Convexity Results for Queueing Systems. Operations Research, Vol. 35, No. 3, 405-418.
%D 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.
%D 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.
%D 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.
%D Medhi, J. 2003. Stochastic Models in Queueing Theory. Second ed., Academic Press, New York.
%D 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.
%H Alois P. Heinz, <a href="/A122525/b122525.txt">Rows n = 1..141, flattened</a>
%H E. Brockmeyer, H. L. Halstrøm and Arne Jensen, <a href="https://web.archive.org/web/20120207182954/http://oldwww.com.dtu.dk/teletraffic/Erlang.html">The Life and Works of A.K. Erlang</a>
%F An equivalent expression for G(s, rho) that is often used is: G(s, rho) = (1-rho)*(s-1)!*Sum_{i=0..s-1} (s^i*rho^i/i!) + rho^s*s^(s-1).
%F 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).
%F For s > 0 and rho > 0 one can also use the integral representation G(s, rho) = ((s*rho)^s/s)*Integral_{t=0..infinity} (rho*s*exp(-rho*s*t)*t*(1+t)^(s-1) dt.
%F Multiplying the n-th row entries by n+1 results in triangle A066324 in row reversed form. - _Peter Bala_, Sep 30 2011
%F 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
%e 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 4, 3;
%e 6, 18, 24, 16;
%e 24, 96, 180, 200, 125;
%e 120, 600, 1440, 2160, 2160, 1296;
%e 720, 4320, 12600, 23520, 30870, 28812, 16807;
%e 5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144;
%e 40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969;
%p G:= proc(s) G(s):= (s-1)!/s*add((s-i)/i!*(s*rho)^i, i=0..(s-1)) end:
%p T:= n-> coeff(G(n), rho, k):
%p seq(seq(T(n, k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Sep 08 2012
%t (* First program *)
%t nn=6; t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; f[list_]:=Select[list,#>0&]; Map[f,Map[Reverse,Range[0,nn]!CoefficientList[Series[Exp[t]/(1-y t),{x,0,nn}],{x,y}]]]//Grid (* _Geoffrey Critzer_, Sep 08 2012 *)
%t (* Second program *)
%t T[n_, k_]:= Coefficient[Series[((n-1)!/n)*Sum[(n-j)*(n*x)^j/j!, {j,0,n-1}], {x,0,30}], x, k];
%t Table[T[n, k], {n,10}, {k,0,n-1}]//Flatten (* _G. C. Greubel_, Jan 06 2022 *)
%o (Sage)
%o def A122525(n,k): return ( (factorial(n-1)/n)*sum((n-j)*(n*x)^j/factorial(j) for j in (0..n-1)) ).series(x, n+1).list()[k]
%o flatten([[A122525(n,k) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Jan 06 2022
%Y Cf. A066324, A137216, A137227.
%K nonn,tabl
%O 1,4
%A _Arie Harel_, Sep 14 2006
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