

A122525


Triangle read by rows: G(s,rho) = ((s1)!/s)*Sum(((si)/i!)*(s*rho)^i, i=0..(s1)).


5



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
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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 wellknown 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 steadystate 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,nk) is the number of partial functions on {1,2,...,n} with exactly k recurrent elements for 0<=k<=n. Row sums = (n+1)^n.  From 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, 959972.
Harel, A. and P. Zipkin. 1987a. Strong Convexity Results for Queueing Systems. Operations Research, Vol. 35, No. 3, 405418.
Harel, A. and P. Zipkin. 1987b. The Convexity of a General Performance Measure for the MultiServer Queues. Journal of Applied Probability, Vol. 24, 725736.
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), 281282.
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, 920923.
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, 3955.


LINKS

Alois P. Heinz, Rows n = 1..141, flattened
Erlang, A. K. Erlang


FORMULA

An equivalent expression for G(s,rho) that is often used is: G(s,rho)=Sum(s^i*rho^i/i!,i=0..s1)*(1rho)*(s1)!+rho^s*s^(s1);
For s>0 and rho>0 one can use the expression: G(s,rho)=(exp(s*rho)*s*rho*(1rho)*(s1)*GAMMA(s1,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)^(s1), t=0..infinity);
Multiplying the nth 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/(1x*t)) evaluated at x = 0, where D is the operator exp(x)/(1x)*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):= (s1)!/s*add((si)/i!*(s*rho)^i, i=0..(s1)) end:
T:= n> coeff(G(n), rho, k):
seq(seq(T(n, k), k=0..n1), n=1..10); # Alois P. Heinz, Sep 08 2012


MATHEMATICA

nn=6; t=Sum[n^(n1)x^n/n!, {n, 1, nn}]; f[list_]:=Select[list, #>0&]; Map[f, Map[Reverse, Range[0, nn]!CoefficientList[Series[Exp[t]/(1y t), {x, 0, nn}], {x, y}]]]//Grid (* Geoffrey Critzer, Sep 08 2012 *)


CROSSREFS

Cf. A066324.
Sequence in context: A329902 A091274 A330745 * A054589 A051851 A336165
Adjacent sequences: A122522 A122523 A122524 * A122526 A122527 A122528


KEYWORD

nonn,tabl


AUTHOR

Arie Harel, Sep 14 2006


STATUS

approved



