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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.

References

  • Benjaafar, S., & Diwakar, G. (1999). Workload allocation in multi-product, multi-facility production systems with setup times. IIE Transactions, 31, 339–352.

    Google Scholar 

  • Calabrese, J. K. (1992). Optimal workload allocation in open networks of multiserver queues. Management Science, 38(12), 1792–1802.

    Article  Google Scholar 

  • Cheah, J., & Smith, J. M. (1994). Generalized M/G/c/c state dependent queueing models and pedestrian traffic flows. Questa, 15, 365–386.

    Google Scholar 

  • Dallery, Y., & Stecke, K. E. (1990). On the optimal allocation of servers and workloads in closed queueing networks. Operations Research, 38(4), 694–703.

    Article  Google Scholar 

  • Harel, A. (2011). Convexity results for the Erlang delay and loss formulae when the server utilization is held constant. Operations Research, 59(6), 1420–1426.

    Article  Google Scholar 

  • Harel, A., & Zipkin, P. H. (1987). Strong convexity results for queueing systems. Operations Research, 35(3), 405–418.

    Article  Google Scholar 

  • Heyman, D. P., & Sobel, M. J. (1982). Stochastic models in operations research: vol. I. New York: Dover.

    Google Scholar 

  • Hillier, F., & So, K. (1995). On the optimal design of tandem queueing systems with finite buffers. Queueing Systems, 21, 245–266.

    Article  Google Scholar 

  • IMSL Software Systems (1987). MATH/Library, Fortran subroutines for mathematical applications.

  • Kelton, D., Sadowski, R., & Swets, N. (2010). Simulation with arena, 5th ed. New York: McGraw-Hill.

    Google Scholar 

  • Kleinrock, L. (1964). Communication nets: stochastic message flow and delay. New York: McGraw-Hill (Reprinted by Dover Publications, 1972).

    Google Scholar 

  • Kleinrock, L. (2002). Creating a mathematical theory of computer networks. Operations Research, 50(2), 125–131.

    Article  Google Scholar 

  • Kobayashi, H., & Mark, B. L. (2002). Generalized loss models and queueing-loss networks. International Transactions in Operational Research, 9, 97–112.

    Article  Google Scholar 

  • Kobayashi, H., & Mark, B. L. (2009). System modeling and analysis: foundations of system performance evaluation. Upper Saddle River: Pearson.

    Google Scholar 

  • Lee, H. F., Srinivasan, M. M., & Yano, C. (1988). Some characteristics of optimal workload allocation for closed queueing networks. Dept. of Industrial and Operations Eng., University of Michigan, Ann Arbor, MI.

  • Reiser, M., & Lavenberg, S. (1980). Mean value analysis of closed multi-chain queueing networks. Journal of the ACM, 27(2), 313–322.

    Article  Google Scholar 

  • Ross, S. (2003). Introduction to probability models (8th ed.). San Diego: Academic Press.

    Google Scholar 

  • Schittkowski, K. (1985/86). NLPQL: a FORTRAN subroutine solving constrained nonlinear programming problems. Annals of Operations Research, 5, 485–500.

    Article  Google Scholar 

  • Schweitzer, P., & Seidmann, A. (1991). Optimizing processing rates for flexible manufacturing systems. Management Science, 37(4), 454–466.

    Article  Google Scholar 

  • Shantikumar, J. G., & Yao, D. (1986). Optimal server allocation in a system of multiserver stations. Management Science, 33, 1173–1180.

    Article  Google Scholar 

  • Smith, J. M. (2010). Robustness of state-dependent queues and material handling systems. Int. J. Prod. Res., 48(16), 4631–4663.

    Article  Google Scholar 

  • Smith, J. M., & Kerbache, L. (2012). State-dependent models of material handling systems in closed queueing networks. Int. J. Prod. Res., 50(2), 461–484.

    Article  Google Scholar 

  • Smith, J. M., & Towsley, D. (1982). The use of queueing networks in the evaluation of egress from buildings. Environment and Planning B, 8, 125–139.

    Article  Google Scholar 

  • Spinellis, D., Papadopoulos, C., & Smith, J. M. (2000). Large production line optimization using simulated annealing. International Journal of Production Research, 38(3), 509–541.

    Article  Google Scholar 

  • Stecke, K. E. (1983). Formulation and solution of nonlinear integer production planning problems for flexible manufacturing systems. Management Science, 29(3), 273–288.

    Article  Google Scholar 

  • Stecke, K. E. (1986). A hierarchical approach to solving machine grouping and loading problems of flexible manufacturing systems. European Journal of Operational Research, 24, 369–378.

    Article  Google Scholar 

  • Stecke, K. E., & Solberg, J. J. (1985). The optimality of unbalancing both workloads and machine group sizes in CLosed queueing networks. Operations Research, 33(4), 882–910.

    Article  Google Scholar 

  • Sung, S. T. (2005). On the optimal workloads allocation of an FMS with finite in-process buffers. In O. Gervasi et al. (Eds.), LNCS: Vol. 3483. ICSA 2005 (pp. 624–631).

    Google Scholar 

  • Sung, C. S., & Kwon, S. T. (1994). Performance modeling of an FMS with finite input and output buffers. International Journal of Production Economics, 37, 161–175.

    Article  Google Scholar 

  • Tregenza, P. (1976). The design of interior circulation. New York: Van Nostrand Reinhold.

    Google Scholar 

  • Yao, D. (1985). Some properties of the throughput function of closed networks of queues. Operations Research Letters, 3(6), 313–317.

    Article  Google Scholar 

Download references

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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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