ABSTRACT
We investigate a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. In general, the optimal policy has a complex dependency on all the input parameters and state information. Our main contribution is to derive index policies that can serve as heuristics and are shown to give good performance. Our index policy attributes to each class an index, which depends on the number of customers currently present in that class. The index values are obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory. They can be expressed as a function of the steady-state distribution probabilities of a one-dimensional birth-and-death process. For linear holding cost, the index can be calculated in closed-form and turns out to be independent of the arrival rates and the number of customers present. In the case of no abandonments and linear holding cost, our index coincides with the cμ-rule, which is known to be optimal in this simple setting. For general convex holding cost we derive properties of the index value in limiting regimes: we consider the behavior of the index (i) as the number of customers in a class grows large, which allows us to derive the asymptotic structure of the index policies, and (ii) as the abandonment rate vanishes, which allows us to retrieve an index policy proposed for the multi-class M/M/1 queue with convex holding cost and no abandonments. In fact, in a multi-server environment it follows from recent advances that the index policy is asymptotically optimal for linear holding cost. To obtain further insights into the index policy, we consider the fluid version of the relaxed problem and derive a closed-form expression for the fluid index. The latter coincides with the stochastic model in case of linear holding costs. For arbitrary convex holding cost the fluid index can be seen as the Gcμθ-rule, that is, including abandonments into the generalized cμ-rule (Gcμ-rule). Numerical experiments show that our index policies become optimal as the load in the system increases.
- P. S. Ansell, K. D. Glazebrook, J. Ni no-Mora, and M. O'Keeffe. Whittle's index policy for a multi-class queueing system with convex holding costs. Mathematical Methods of Operations Research, 57(1):21--39, 2003.Google ScholarCross Ref
- N.T. Argon, S. Ziya, and R. Righter. Scheduling impatient jobs in a clearing system with insights on patient triage in mass-casualty incidents. Prob. Eng. Inf. Sci., 22(3):301--332, 2010. Google ScholarDigital Library
- B. Ata and M.H. Tongarlak. On scheduling a multiclass queue with abandonments under general delay costs. Queueing Systems, 74:65--104, 2013. Google ScholarDigital Library
- R. Atar, C. Giat, and N. Shimkin. The cμ/θ rule for many-server queues with abandonment. Operation Research, 58(5):1427--1439, 2010. Google ScholarDigital Library
- R. Atar, C. Giat, and N. Shimkin. On the asymptotic optimality of the cμ/θ rule under ergodic cost. Queueing Systems, 67:127--144, 2011. Google ScholarDigital Library
- F. Avram, D. Bertsimas, and M. Richard. Optimization of multiclass queuing networks: a linear control approach. Stochastic Networks, eds. F.P. Kelly and R.J. Williams, pages 199--234, 1995.Google Scholar
- U. Ayesta, P. Jacko, and V. Novak. A nearly-optimal index rule for scheduling of users with abandonment. In IEEE Infocom 2011, 2011.Google Scholar
- N. Bäuerle. Asymptotic optimality of tracking policies in stochastic networks. Ann. Appl. Probab, 10:1065--1083, 2000.Google ScholarCross Ref
- N. Bäuerle and U. Rieder. Optimal control of singleserver fluid networks. Queueing Syst., 35:185--200, 2000. Google ScholarDigital Library
- S. Bhulai, H. Blok, and F.M. Spieksma. k computing queues with customer abandonment: optimality of a generalized cμ-rule by the smoothed rate truncation method. Work in progress, 2014.Google Scholar
- S. Bhulai, A.C. Brooms, and F.M. Spieksma. On structural properties of the value function for an unbounded jump markov process with an application to a processor sharing retrial queue. QUESTA, 76:425--446, 2014. Google ScholarDigital Library
- C.F. Bispo. The single-server scheduling problem with convex costs. Queueing Systems, 73:261--294, 2013. Google ScholarDigital Library
- C. Buyukkoc, P. Varaya, and J. Walrand. The cη rule revisited. Adv. Appl. Prob., 17:237--238, 1985.Google ScholarCross Ref
- J.G. Dai. On positive harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Annals of Applied Probability, 5:49--77, 1995.Google ScholarCross Ref
- J.G. Dai and S. He. Many-server queues with customer abandonment: A survey of diffusion and fluid approximations. J. Syst. Sci. Syst. Eng., 21:1--36, 2012.Google ScholarCross Ref
- D.G. Down, G. Koole, and M.E. Lewis. Dynamic control of a single server system with abandonments. Queueing Systems, 67:63--90, 2011. Google ScholarDigital Library
- A. Gajrat and A. Hordijk. Fluid approximation of a controlled multiclass tandem network. Queueing Systems, 35:349--380, 2000. Google ScholarDigital Library
- E. Gelenbe and I. Mitrani. Analysis and Synthesis of Computer Systems. London: Academic Press, 1980.Google Scholar
- J.C. Gittins, K. Glazebrook, and R. Weber. Multiarmed Bandit Allocation Indices. Wiley, 2011.Google ScholarCross Ref
- K.D. Glazebrook, P.S. Ansell, R.T. Dunn, and R.R. Lumley. On the optimal allocation of service to impatient tasks. J. Appl. Prob., 41:51--72, 2004.Google ScholarCross Ref
- K.D. Glazebrook, D.J. Hodge, C. Kirkbride, and R.J. Minty. Stochastic scheduling: A short history of index policies and new approaches to index generation for dynamic resource allocation. J. of Sched., 2013.Google Scholar
- K.D. Glazebrook, C. Kirkbride, and J. Ouenniche. Index policies for the admission control and routing of impatient customers to heterogeneous servcie stations. Operations Research, 57:975--989, 2009. Google ScholarDigital Library
- J. Hasenbein and D. Perry (Eds). Special issue on queueing systems with abandonments. Queueng Systems, 75(2-4):111--384, 2013. Google ScholarDigital Library
- J. Kim and A. Ward. Dynamic scheduling of a GI/GI/1+GI queue with multiple customer classes. Queueing Systems, 75(2--4):339--384, 2013. Google ScholarDigital Library
- M. Larrañaga, U. Ayesta, and I.M. Verloop. Index policies for a multi-class queue with convex holding cost and abandonments. LAAS technical report 14098.Google Scholar
- M. Larrañaga, U. Ayesta, and I.M. Verloop. Dynamic fluid-based scheduling in a multi-class abandonment queue. Performance Evaluation, 70:841--858, 2013. Google ScholarDigital Library
- A. Mandelbaum and S. Stolyar. Scheduling exible servers with convex delay costs: Heavy-traffic optimality of the generalized cμ-rule. Operations Research, 52(6):836--855, 2004. Google ScholarDigital Library
- J. Ni no-Mora. Dynamic priority allocation via restless bandit marginal productivity indices. TOP, 15(2):161--198, 2007.Google ScholarCross Ref
- M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, 2005.Google ScholarDigital Library
- J. A. van Mieghem. Dynamic scheduling with convex delay costs: The generalized cμ rule. The Annals of Applied Probability, 5(3):808--833, 1995.Google ScholarCross Ref
- I.M. Verloop. Asymptotic optimal control of multiclass restless bandits. CNRS Technical Report, hal-00743781, September 2013.Google Scholar
- I.M. Verloop and R. Núñez-Queija. Asymptotically optimal parallel resource assignment with interference. Queueing Systems, 65(1):43--92, 2010. Google ScholarDigital Library
- R.R. Weber and G. Weiss. On an index policy for restless bandits. J. Appl. Prob., 27:637--648, 1990.Google ScholarCross Ref
- G. Weiss. On optimal draining of reentrant fluid lines. Stochastic Networks, eds. F.P. Kelly and R.J. Williams, pages 91--103, 1995.Google Scholar
- P. Whittle. Restless bandits: Activity allocation in a changing world. J. Appl. Prob., 25:287--298, 1988.Google ScholarCross Ref
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- Index policies for a multi-class queue with convex holding cost and abandonments
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