OFFSET
1,4
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.
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
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.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
E. Brockmeyer, H. L. Halstrøm and Arne Jensen, The Life and Works of A.K. Erlang
FORMULA
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).
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)*Integral_{t=0..infinity} (rho*s*exp(-rho*s*t)*t*(1+t)^(s-1) dt.
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
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;
MAPLE
G:= proc(s) G(s):= (s-1)!/s*add((s-i)/i!*(s*rho)^i, i=0..(s-1)) end:
T:= n-> coeff(G(n), rho, k):
seq(seq(T(n, k), k=0..n-1), n=1..10); # Alois P. Heinz, Sep 08 2012
MATHEMATICA
(* First program *)
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 *)
(* Second program *)
T[n_, k_]:= Coefficient[Series[((n-1)!/n)*Sum[(n-j)*(n*x)^j/j!, {j, 0, n-1}], {x, 0, 30}], x, k];
Table[T[n, k], {n, 10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Jan 06 2022 *)
PROG
(Sage)
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]
flatten([[A122525(n, k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Jan 06 2022
KEYWORD
nonn,tabl
AUTHOR
Arie Harel, Sep 14 2006
STATUS
approved