Arie Harel
Abstract
We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We derive simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n , we show that the variance of the total penalty costs of the EDD is linear in n . The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)).
- {1} K. R. Baker and G. D. Scudder, Sequencing with earliness and tardiness penalties: A review, Oper. Res. 38 (1990) 22-36. Google Scholar
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- {2} D. J. Bertsimas and L. D. Servi, Deducing queueing from transactional data: the queue inference engine, revisited, Oper. Res. 40(2) (1992) 217-228. Google ScholarDigital Library
- {3} D. J. Daley and L. D. Servi, Exploiting Markov chains to infer queue length from transactional data, J. Appl. Probab. 29 (1992) 713-732.Google ScholarCross Ref
- {4} D. J. Daley and L. D. Servi, Estimating waiting times from transactional data, OR Journal on Computing 9 (1997) 224-229.Google Scholar
- {5} D. J. Daley and L. D. Servi, Approximating last-exit probabilities of a random walk, with applications to conditional queue length moments within busy periods of M/GI /1 queues, J. Appl. Probab. 31A (1994) 251-267.Google ScholarCross Ref
- {6} J. Du and J. Y.-T. Leung, Minimizing total tardiness on one machine is NP-hard, Math. Oper. Res. 15 (1990) 483-495. Google ScholarDigital Library
- {7} T. D. Fry, R. D. Armstrong and J. H. Blackstone, Minimizing weighted absolute deviation in single machine scheduling, IIE Trans. 19 (1987) 445-450.Google Scholar
- {8} M. R. Garey, R. E. Tarjan and G. T. Wilfong, One-processor scheduling with symmetric earliness and tardiness penalties, Math. Oper. Res. 13 (1988) 330-348. Google ScholarDigital Library
- {9} H. W. Gould, Combinatorial identities: A standardized set of tables listing 500 binomial coefficient summations, published by the author (1972).Google Scholar
- {10} N. G. Hall, W. Kubiak and S. P. Sethi, Earliness-tardiness scheduling problem, II: Deviation of completion times about a restrictive common due date, Oper. Res. 39 (1991) 847-856. Google ScholarDigital Library
- {11} N. G. Hall and M. E. Posner, Earliness-tardiness scheduling problems, I: Weighted deviation of completion times about a common due date, Oper. Res. 39 (1991) 836-846. Google ScholarDigital Library
- {12} A. Harel, On honest Bernoulli excursions, J. Appl. Probab. 27 (1990) 475-476.Google ScholarCross Ref
- {13} A. Harel, Random walk and the area underneath its path, Math. Oper. Res. 18(3) (1993) 566-577. Google ScholarDigital Library
- {14} L. K. Jones and R. C. Larson, Efficient computation of probabilities of events described by order statistics and applications to queue inference, ORSA J. Comput. 7(1) (1995) 89-100.Google Scholar
- {15} S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes (Academic Press, 1981).Google Scholar
- {16} R. C. Larson, The queue inference engine: deducing queue statistics from transactional data, Management Sci. 36 (1990) 586-601. Google ScholarDigital Library
- {17} R. C. Larson, The queue inference engine: Addendum, Management Sci. 37(8) (1991) 1062. Google ScholarDigital Library
- {18} P. S. Ow and T. E. Morton, The single machine early-tardy problem, Management Sci. 35 (1989) 177-191. Google ScholarDigital Library
Index Terms
- Order statistics applications to queueing and scheduling problems
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