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Optimal workload allocation in closed queueing networks with state dependent queues

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Abstract

The problem of optimal workload allocation in closed queueing network models with multi-server exponential infinite capacity workstations and finite capacity state dependent queueing models is examined. The processing rates (i.e. service times) of jobs in the queueing system are the main focus. State dependent queues are appropriate for modeling the transportation and material transfer of the customers and products within the system. By combining these two types of queues, we can model the processing of jobs or customers at service stations and also the material handling transfer of job and customers between these stations. Because the environment is a closed queueing network model, it allows for the dynamic interaction of the jobs and customers in the optimal workload allocation values. This combination of queues and optimal search process provides a broad range of potential applications in manufacturing and service type design operations where the travel time between workstations is important. Conveyors, automated-guided vehicle systems (AGVS), and fork lift trucks are utilized to show the full range of material handling component systems.

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Abbreviations

A :

The number of queueing chains (e.g. separate population paths) in the network.

α ij :

Generic routing probability from node i to j in the network.

c :

Number of servers in the queues.

d ℓ :

Cost per unit of service rate at node ℓ.

\(d_{V_{1}}^{\ell}\) :

Cost per unit of service rate at the M/G/c/c node.

D :

Right-hand-side upper bound of resource available in knapsack constraint.

G :

General service time distribution.

\(\mathcal{G}\) :

Normalization constant.

G(V,E):

The graph topology of the closed queueing network with a finite set of nodes V and a finite set of edges E.

k :

Index on a queueing chain in the network.

λ ℓk :

Poisson arrival rate to node ℓ in chain k.

\(\mathcal{L}\) :

Length in meters of an M/G/c/c queue.

M :

Poisson arrival process (i.e. Markovian).

μ ℓ :

Exponential mean service rate at node ℓ.

μ c :

The processing rate for an M/G/c/c queue with fixed length and width (i.e. μ c =f(V 1,L,W) free-flow speed V 1, capacity c∼f(length×width).

n :

Number of customers in a queue.

N :

Number of stations in the network.

P n :

Probability of n customers in a queue.

π ℓ :

Stationary probability of the number of customers at queue ℓ in a closed queueing network.

\(\rho=\frac{\lambda}{c\mu }\) :

Proportion of time each server is busy.

θ :

Throughput rate.

\(\mathcal{W}\) :

Width in meters of an M/G/c/c queue.

θ(W k ):

Throughput of the closed queueing network as a function of the finite population W k .

V 1 :

Free flow speed service rate parameter/variable of the M/G/c/c queue.

W k :

The population of products(customers) in a single chain(class).

w ℓk :

Average delay at node ℓ in closed network.

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  1. Department of Mechanical and Industrial Engineering, University of Massachusetts Amherst, Amherst, MA, 01003, USA

    J. MacGregor Smith

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Smith, J.M. Optimal workload allocation in closed queueing networks with state dependent queues. Ann Oper Res 231, 157–183 (2015). https://doi.org/10.1007/s10479-013-1418-0

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