Function File: [U, R, Q, X] = qnmix (lambda, N, S, V, m)

Mean Value Analysis for mixed queueing networks. The network consists of K service centers (single-server or delay centers) and C independent customer chains. Both open and closed chains are possible. lambda is the vector of per-chain arrival rates (open classes); N is the vector of populations for closed chains.

Class switching is not allowed. Each customer class must correspond to an independent chain.

If the network is made of open or closed classes only, then this function calls qnom or qncmmva respectively, and prints a warning message.

INPUTS

lambda(c)
N(c)

For each customer chain c:

  • if c is a closed chain, then N(c)>0 is the number of class c requests and lambda(c) must be zero;
  • If c is an open chain, lambda(c)>0 is the arrival rate of class c requests and N(c) must be zero;

In other words, for each class c the following must hold: 

(lambda(c)>0 && N(c)==0) ||(lambda(c)==0 && N(c)>0)
S(c,k)

mean class c service time at center k, S(c,k) ≥ 0. For FCFS nodes, service times must be class-independent.

V(c,k)

average number of visits of class c customers to center k (V(c,k) ≥ 0).

m(k)

number of servers at center k. Only single-server (m(k)==1) or IS (Infinite Server) nodes (m(k)<1) are supported. If omitted, each center is assumed to be of type M/M/1-FCFS. Queueing discipline for single-server nodes can be FCFS, PS or LCFS-PR.

OUTPUTS

U(c,k)

class c utilization at center k.

R(c,k)

class c response time at center k.

Q(c,k)

average number of class c requests at center k.

X(c,k)

class c throughput at center k.

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 7.4.3 ("Mixed Model Solution Techniques"). Note that in this function we compute the mean response time R instead of the mean residence time as in the reference.
  • Herb Schwetman, Implementing the Mean Value Algorithm for the Solution of Queueing Network Models, Technical Report CSD-TR-355, Department of Computer Sciences, Purdue University, revised Feb 15, 1982.

See also: qncmmva, qncm.

Package: queueing