INPUTS

lambda(c)

If this function is invoked as qnom(lambda, S, V, …), then lambda(c) is the external arrival rate of class $c$ customers (lambda(c) ≥ 0). If this function is invoked as qnom(lambda, S, P, …), then lambda(c,k) is the external arrival rate of class $c$ customers at center $k$ (lambda(c,k) ≥ 0).

S(c,k)

mean service time of class $c$ customers on the service center $k$ (S(c,k)>0). For FCFS nodes, mean service times must be class-independent.

V(c,k)

visit ratio of class $c$ customers to service center $k$ (V(c,k) ≥ 0 ). If you pass this argument, class switching is not allowed

P(r,i,s,j)

probability that a class $r$ job completing service at center $i$ is routed to center $j$ as a class $s$ job. If you pass argument P, class switching is allowed; however, all servers must be fixed-rate or infinite-server nodes (m(k) ≤ 1 for all $k$).

m(k)

number of servers at center $k$. If m(k) < 1, enter $k$ is a delay center (IS); otherwise it is a regular queueing center with m(k) servers. Default is m(k) = 1 for all $k$.

OUTPUTS

U(c,k)

If $k$ is a queueing center, then U(c,k) is the class $c$ utilization of center $k$. If $k$ is an IS node, then U(c,k) is the class $c$ $traffic\; intensity$ defined as X(c,k)*S(c,k).

R(c,k)

class $c$ response time at center $k$. The system response time for class $c$ requests can be computed as dot(R, V, 2).

Q(c,k)

average number of class $c$ requests at center $k$. The average number of class $c$ requests in the system Qc can be computed as Qc = sum(Q, 2)

X(c,k)

class $c$ throughput at center $k$.

NOTES

If the function call specifies the visit ratios V, class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. Note that the meaning of parameter lambda is different from one case to the other (see below).

REFERENCES

• Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.1 ("Open Model Solution Techniques").

See also: qnopen, qnos, qnomvisits

 P = zeros(2,2,2,2);
 lambda = zeros(2,2);
 S = zeros(2,2);
 P(1,1,2,1) = P(1,2,2,1) = 0.2;
 P(1,1,2,2) = P(2,2,2,2) = 0.8;
 S(1,1) = S(1,2) = 0.1;
 S(2,1) = S(2,2) = 0.05;
 rr = 1:100;
 Xk = zeros(2,length(rr));
 for r=rr
   lambda(1,1) = lambda(1,2) = 1/r;
   [U R Q X] = qnom(lambda,S,P);
   Xk(:,r) = sum(X,1)';
 endfor
 plot(rr,Xk(1,:),";Server 1;","linewidth",2, ...
      rr,Xk(2,:),";Server 2;","linewidth",2);
 legend("boxoff");
 xlabel("Class 1 interarrival time");
 ylabel("Throughput");