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A122525 Triangle read by rows: G(s,rho) = ((s-1)!/s)*Sum(((s-i)/i!)*(s*rho)^i, i=0..(s-1)). 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 (list; table; graph; refs; listen; history; text; internal format)
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. - 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, 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

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..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

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

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 *)

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

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)