Function File: [U, R, Q, X, G] = qncsconvld (N, S, V)

Convolution algorithm for product-form, single-class closed queueing networks with K general load-dependent service centers.

This function computes steady-state performance measures for single-class, closed networks with load-dependent service centers using the convolution algorithm; the normalization constants are also computed. The normalization constants are returned as vector G=[G(1), …, G(N+1)] where G(i+1) is the value of G(i).

INPUTS

N

Number of requests in the system (N>0).

S(k,n)

mean service time at center k where there are n requests, 1 ≤ n ≤ N. S(k,n) = 1 / \mu_{k,n}, where \mu_{k,n} is the service rate of center k when there are n requests.

V(k)

visit count of service center k (V(k) ≥ 0). The length of V is the number of servers K in the network.

OUTPUT

U(k)

center k utilization.

R(k)

average response time at center k.

Q(k)

average number of requests in center k.

X(k)

center k throughput.

G(n)

Normalization constants (vector). G(n+1) corresponds to G(n), as array indexes in Octave start from 1.

REFERENCES

• Herb Schwetman, Some Computational Aspects of Queueing Network Models, Technical Report CSD-TR-354, Department of Computer Sciences, Purdue University, February 1981 (revised).
• M. Reiser, H. Kobayashi, On The Convolution Algorithm for Separable Queueing Networks, In Proceedings of the 1976 ACM SIGMETRICS Conference on Computer Performance Modeling Measurement and Evaluation (Cambridge, Massachusetts, United States, March 29–31, 1976). SIGMETRICS ’76. ACM, New York, NY, pp. 109–117. 10.1145/800200.806187

This implementation is based on G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, pp. 313–317. Function qncsconvld is slightly different from the version described in Bolch et al. because it supports general load-dependent centers (while the version in the book does not). The modification is in the definition of function F() in qncsconvld which has been made similar to function f_i defined in Schwetman, Some Computational Aspects of Queueing Network Models.

See also: qncsconv.

Package: queueing