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**Stochastic monotonicity of the queue lengths in closed queueing networks.**
*(English)*
Zbl 0652.60101

The present paper deals with a closed queueing network with M stations and N jobs (multiple stations and multiple jobs, like the ones in manufacturing and computer systems). Not only is the equilibrium behavior of the system at infinite time discussed but also the transient behavior which is more useful in real situations, introducing the concepts of stochastic order relation and likelihood ratio order relation. There are two major results:

1) Increasing N will stochastically increase the queue-length vector process \((X_ 1(t)\), \(X_ 2(t)\),..., \(X_ M(t))\) over any finite time interval [0,t], provided that all stations have nondecreasing service rates, and

2) in the equilibrium setting, increasing N will increase the sum of queueing lengths at any subset of stations in the sense of likelihood ratio ordering if and only if the complement subnetwork has a nondecreasing throughput.

The latter half is devoted to the proofs of theorems.

1) Increasing N will stochastically increase the queue-length vector process \((X_ 1(t)\), \(X_ 2(t)\),..., \(X_ M(t))\) over any finite time interval [0,t], provided that all stations have nondecreasing service rates, and

2) in the equilibrium setting, increasing N will increase the sum of queueing lengths at any subset of stations in the sense of likelihood ratio ordering if and only if the complement subnetwork has a nondecreasing throughput.

The latter half is devoted to the proofs of theorems.

Reviewer: Y.Sakai

### MSC:

60K25 | Queueing theory (aspects of probability theory) |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

90B22 | Queues and service in operations research |