Published on: Wed, 03/29/2017

Last Modified: Thu, 01/23/2020 - 11:16am

Last Modified: Thu, 01/23/2020 - 11:16am

__LIST OF PUBLICATIONS__

A. __REFEREED ARTICLES__

__ARTICLES IN REFEREED JOURNALS__

- Brill, P.H., Huang, M.L., Hlymka, M. (2019). "On the Service Time in a Workload-barrier M/G/1 Queue with Accepted and Blocked Customers", European Journal of Operational Research, available online 26 October 2019; Volume 283, Issue 1, 16 May 2020, Pages 235-243.

- Zhang, Y., Hlynka, M., Brill, P.H. (2019). “First Passage and Collective Marks” International Journal of Statistics and Probability; Vol. 8, No. 6; November 2019. ISSN 1927-7032 E-ISSN 1927-7040. Published by Canadian Center of Science and Education.

- Brill, P. H., Cheung, Chi Ho, Hlynka, M., Jiang, Q. (2018), “Reversibility Checking for Markov Chains", Communications on Stochastic Analysis, Vol. 12, No. 2 , 129-135.

- Brill, P. H. (2015). “Note on the Service Time in M/G/1 Queues with Bounded Workload”, Statistics and Probability Letters, Vol. 96, January 2015, Pages 162-169. [28(2), 1–19 doi:10.1017/S0269964813000417.]

- Brill, P. H. (2014). “Alternative Analysis of Finite-Time Probability Distributions of Renewal Theory”, Probability in the Engineering and Informational Sciences”, Volume 28, Issue 2 April 2014, pp. 183-201. [28(2), 1–19 doi:10.1017/S0269964813000417.]

- Huang, M. L., Coia, V., Brill, P. H. (2013). "A Cluster Truncated Pareto Distribution and its Applications", ISRN Probability and Statistics, Volume 2013 (2013), Article ID 265373, 10 pages

- Brill, P. H., Hlynka, M. (2012). “Server Workload in an M/M/1 Queue with Bulk Arrivals and Special Delays”, Applied Mathematics, Vol. 3, No. 12A (Special issue on Probability and Its Applications, December 2012, PP. 2174-2177. Paper invited by editor-in-chief, Chris Cannings, Univ. of Sheffield, U.K. Published Online December 2012 (http://www.SciRP.org/journal/am).

- Yu, K., Huang, M. L., Brill, P. H. (2012). “An Algorithm for Fitting Heavy-tailed Distributions via Generalized Hyper-exponentials”, INFORMS Journal on Computing, 24(1), 42-52.

- Brill, P. H., Yu, K. (2011). “Analysis of Risk Models Using a Level Crossing Technique”, Insurance: Mathematics and Economics, 49(3), 298-309.

- Hlynka, M., Brill, P. H., Horn, W. (2010). “A Method for Obtaining Laplace Transforms of Order Statistics of Erlang Random variables”, Statistics and Probability Letters 80, 9-18.

- Brill, P. H., Huang, M. L., Hlynka, M. (2009). “Note on an <s, S> Inventory System with Decay”, IAENG International Journal of Applied Mathematics, 39(3), 171-174.

- Brill, P. H. (2009). “Compound Cycle of a Renewal Process and Applications”, INFOR (Canadian Operational Res. Society) Special Issue on Queues, 47(4), 273-281.

- Brill, P. H. (2009). "Note on a Series for M/G/1 Queues", International J. of Operational Research (IJOR), 5(3), 363-373.

- Mandelbaum, M., Hlynka, M., Brill, P. H. (2007) “Nonhomogeneous Geometric Distribution with Relations to Birth and Death Processes”, J. of the Spanish Soc. of Statistics and Operations Research (TOP), 15(2), 281-296, Cover Date: 2007-12-01, ISSN: 1134-5764.

- Shortle, J. F., Fischer, M., Brill, P. H. (2007). "Waiting Time Distribution of M/DN/1 Queues Through Numerical Laplace Inversion", INFORMS J. on Computing, 19(1), 112-120.

- Wu, X., Brill, P. H., Hlynka, M., Wang, J. (2005). “An M/G/1 Retrial Queue with Balking and Retrials During Service”, International J. of Operational Research (IJIOR), 1(1-2), 30-51.

- Shortle, J. F., Brill, P. H. (2005). "Analytical Distribution of Waiting Time in the M/{iD}/1 Queue", Queueing Systems 50(2), 185-197.

- Shortle, J. F., Brill, P. H., Fischer, M. J., Gross, D., Masi, D.M. (2004). “An Algorithm to Compute the Distribution of the waiting Time in the M/G/1 Queue”, IINFORMS J. on Computing, 16(2), 152-161.

- Huang, M. L., Brill, P. H. (2004). “A Level Crossing Distribution Estimation Method”, J. of Statistical Planning and Inference, 124(1), 45-62.

- Brill, P. H., Hlynka, M. (2002). “A Geometric Interpretation of Two-State Markov Transition Matrices”, Missouri J. of Mathematical Sciences, 14(3), 159-174.

- Huang, M. L., Brill, P. H. (2001). “On Estimation in M/G/c/c Queues”, International Transactions in Operational Research, 8(6), 647-657.

- Huang, M. L., Brill, P. H. (2001). “A Nonparametric Regression Method”, Nonlinear Analysis, Theory, Methods and Applications, 47(3), 1467-1475.

- Harris, C. M., Brill, P. H., Fischer, M. (2000). “Internet-type Queues with Power-tailed Interarrival Times and Computational Methods for their Analysis”, INFORMS J. on Computing, 12, 448-458.

- Brill, P. H., Hlynka, M. (2000). “An Exponential Queue with Competition for Service”, European Journal of Operational Research, 126(3), 587-602.

- Huang, M. L., Brill, P. H. (1999). “A Level Crossing Quantile Estimation Method”, Statistics and Probability Letters, 45, 111-119.

- Brill, P. H., Hlynka, M. (1998). “A Single Server N-Line Queue in which a Customer May Receive Special Treatment”, Stochastic Models, 14(4), 905-931.

- Brill, P. H., Harris, C. M. (1997). “M/G/1 Queues with Markov-generated Server Vacations”, Stochastic Models, 13(3), 491-521.

- Huang, M. L., Brill, P. H. (1997). “A Level Crossing Density Estimation Method”, Nonlinear Analysis, Theory, Methods and Applications, 30(7), 4403-4414.

- Brill, P. H., Chaouch, B. A. (1995). “An EOQ Model with Random Variation in Demand”, Management Science, 41(5), 927-936.

- Brill, P. H., Harris, C. M. (1992). “Waiting Time for M/G/1 Queues with Server Time Dependent Server Vacations”, Naval Research Logistics, 39, 775-787.

- Azoury, K., Brill, P. H. (1992). “Analysis of Net Inventory in Continuous Review Models with Random Lead Time”, European Journal of Operational Research, 59, 383-392.

- Huang, M. L., Brill, P. H. (1991). “Recurrence Relations for the Minimum Variance Unbiased Estimator of the Probability Density Function of the R Distribution”, Communications in Statistics Theory and Methods, 20, 4005-4019.

- Brill, P.H., Mandelbaum, M. (1990). “Measurement of Adaptivity and Flexibility in Production Systems”, European Journal of Operational Research, 49, 325-332.

- Mandelbaum. M., Brill, P. H. (1989). “Examples of Measurement of Flexibility and Adaptivity in Production Systems”, Journal of the Operational Research Society, 40, 603-609.

- Brill, P. H., Mandelbaum, M. (1989). “On Measures of Flexibility in Manufacturing Systems”, International Journal of Production Research, 27, 747-756.

- Brill, P. H. (1988). “Single Server Queues with Delay-dependent Arrival Streams”, Probability in the Engineering and Informational Sciences, 2, 231-247.

- Azoury, K., Brill, P. H. (1986). “An Application of the System Point Method to Inventory Models under Continuous Review”, Journal of Applied Probability, 23, 778-789.

- Brill, P. H., Green, L. (1984). “Queues in which Customers Receive Simultaneous Service from a Random Number of Servers: A System-Point Approach”, Management Science, 30(1), 51-68.

- Brill, P. H., Hornik, J. (1984). “System Point Approach to Non-uniform Advertising Insertions”, Operations Research, 32, No. 1, 7-22.

- Brill, P. H., Posner, M. (1981). “A Two-server Queue with Non-waiting Customers Receiving Specialized Service”, Management Science, 8, 914-925.

- Brill, P. H., Posner, M. (1981). “The System Point Method in Exponential Queues: A Level Crossing Approach”, Mathematics of Operations Research, 6, 31-49.

- Brill, P. H., Moon, R.F. (1980). “An Application of Queueing Theory to Pharmacokinetics”, Journal of Pharmaceutical Sciences, 69, 558-560.

- Brill, P. H. (1979). “An Embedded Level Crossing Technique for Dams and Queues”, Journal of Applied Probability, 16, 174-186.

- Brill, P. H., Posner, M. (1977). “Level Crossings in Point Processes Applied to Queues: Single Server Case”, Operations Research, 23, 662-673.

__REFEREED ARTICLE IN A CORPORATE JOURNAL__

- Fischer, M. J., Masi, D.M.B, Brill, P.H., Gross, D., Shortle, J. F. (2002). “Structural Properties of the Transform Approximation Method and Recursion Method”, Telecommunications Review, (13), 121-128. Mitretek Systems (now NOBLIS), Falls Church VA, USA.

__REFEREED ARTICLE IN ENCYCLOPEDIA__

- Brill, P. H. (1996). “Level Crossing Methods”, in Encyclopedia of Operations Research and Management Science, S. Gass and C.M. Harris, Editors. Kluwer Academic Publications, Norwell, Mass., invited article. 338-340.

__REFEREED INVITED ARTICLE IN CORS BULLETIN __

- Brill, P. H. (2000). “A Brief Outline of the Level Crossing Method in Stochastic Models”, English/French, CORS-SCRO (Canadian Operational Res. Society) Bulletin, 34(4), 9-21.

__ABSTRACTS IN REFEREED JOURNALS__

- Brill P. H. (1977). “The System Point Process in Queueing”, Advances in Applied Probability, 9(2), p. 216. Abstract of presentation at the Sixth Conference on Stochastic Processes and their Applications, Tel Aviv, Israel, June, 1976.

- Brill, P. H. (1976). “A New Methodology for Modelling a Broad Class of Exponential Queues”, Advances in Applied Probability, 8(2), p. 242. Abstract of presentation at the Fifth Conference on Stochastic Processes and their Applications, University of Maryland, College Park, MD, USA, June 1975. Invited by Dr. J. Keilson (the external examiner for my PhD thesis) and Dr. R. Syski, conference chair.

__REFEREED ARTICLES IN CONFERENCE PROCEEDINGS __

- Brill, P. H. and Huang, M. L. (2018). “The Posterior Service Time in an M/G/1 Queue with a Workload Barrier and Extreme Prior Service Times”, 2018 JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association, pp. 1051-1056.

- Brill, P. H. and Huang, M. L. (2017). “Approximating Probability Distributions in an Extreme Renewal Process”, 2017 JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association, pp.1395-1401.

- Huang, M. L., *Mottola, J. and Brill, P. H. (2015). “On a Mixture Pareto Distribution”, 2015 JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association, pp. 1416-1427.

- Brill, P. H. and Huang, M. L. (2014). “Example of a Renewal Process with No-mean Interarrival Times”, 2014 JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association, pp. 633-642.

- Huang, M. L., *Thorpe, L., and Brill, P. H. (2013). “A Nonparametric Method for Extreme Values”, 2013 JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association, pp.2905-2914.

- Huang, M. L., Coia, V., Brill, P. H. (2012). “A Mixture Truncated Pareto Distribution”, JSM Proceedings 2012, Statistical Computing Section, Alexandria, VA: American Statistical Association, 2488-2498.

- Huang, M. L., Brill, P. H., Gross, D. (2005). “A Weighted Estimation Method for the Pareto Variance”, 2005 Proceedings of the American Statistical Association, Nonparametric Statistics Section, 1649 -1654.

- Fischer, M. J., Masi, D. M. B., Gross, D., Shortle, J. F., Brill, P.H. (2005). "Development of Procedures to Analyze Queueing Models with Heavy-Tailed Interarrival and Service Times", 2005 NSF Design, Service, and Manufacturing Grantees and Research Conference. January 3-6, 2005. Scottsdale, Ariz.

- Fischer, M.J., Masi, D.M.B., Brill, P.H., Gross, D., Shortle, J. (2004). "Development of Procedures to Analyze Queueing Models with Heavy-Tailed Interarrival and Service Times.” 2004 NSF Design. "Service and Manufacturing Grantees and Research Conference Proceedings (Briefing Slides)." ed. R. Kovacevic. Dallas, Texas, January 5-8, 2004.

- Fischer, M.J., Masi, D.M.B., Gross, D., Shortle, J., Brill, P.H. (2003), “Applying the TAM Recursion Method (TRM) to Analyze an Internat Type Congestion Problem”, Applied Telecommunications Symposium, Orlando, Florida, March 31.

- Fischer, M.J., Masi, D.M.B., Brill, P.H., Gross, D., Shortle, J. (2003), “Development of Procedures to Analyze Queueing Models with Heavy-Tailed Interarrival and Service Times – A Status Report”, 2003 NSF Design, Service and Manufacturing Grantees and Research Conference Proceedings, ed. R.G. Reddy, Birmingham, AL, January 6-9.

- Huang, M.L., Brill, P.H. (2002) “On Rank Tests for Experimental Designs”, American Statistical Association, Proceedings-Statistical Computing Section, New York, 1502-1506.

- Fischer, M.J., Masi, D.M.B., Gross, D., Shortle, J., Brill, P.H. (2001), “Using Quantile Estimates in Simulating Internet Queues with Pareto Service Times”, Proceedings of the 2001 Winter Simulation Conference, ed. B.A. Peters, J.S. Smith, D.J. Medeiros, and M. W. Rohrer, Washington, DC, December, p. 477-485.

- Brill, P. H., Mandelbaum, M. (1999). “Queueing Flexibility in Production Systems”, Proceedings of the 15-th Int. Conference on Production Research, Limerick, Ireland, Eds. M.T. Hillery, H. J. Lewis, University of Limerick, 9th-12th October, 465-467.

- Huang, M.L., Brill, P.H., (1999b) “A Weighted Regression Method”, American Statistical Association, Proceedings of the Statistical Computing Section, Baltimore, 143-156.

- Huang, M.L., Brill, P.H. (1996) “A Nonparametric Quantile Estimation Method”, American Statistical Association, Proceedings of the Statistical Computing Section, 206-211.

- Brill, P.H., Huang, M.L. (1995) “A New Weighted Estimation Method”, Proceedings of the 50th Session of the International Statistical Institute, Beijing, China, 120-121.

- Huang, M.L., Brill, P.H. (1995) “A New Weighted Density Estimation Method”, American Statistical Association, Proceedings of the Statistical Computing Section, 125-130.

- Brill, P. H., Mandelbaum, M. (1995). “Measures of Flexibility in Queueing Systems”, Proceedings of the 13 th Conference on Production Research, Jerusalem, Israel, 518-520.

- Huang, M.L., Brill, P.H. (1994) “Some Estimation Problems in Systems of M/G/c/c Queues”, American Statistical Association, Proceedings of the Science and Engineering Section, 149-154.

- Brill, P.H., Huang, M.L. (1993) “System Point Estimation of the Probability Distribution of the Waiting Time in Variants of M/Ga,b/1 Queues”, American Statistical Association, Proceedings of the Statistical Computing Section 236-241.

- Huang, M.L., Brill, P.H. (1992), “On Estimation of the R Distribution”, American Statistical Association, Proceedings of the Statistical Computing Section, 140-145.

- Brill, P.H. (1991), “Estimation of Stationary Distributions in Storage Processes Using Level Crossing Theory”, American Statistical Association, Proceedings of the Statistical Computing Section, 172-177.

- Huang, M.L., Brill, P.H. (1991), “Recurrence Relations for the Minimum Variance Unbiased Estimator of the Probability Function of the More Generalized Stirling Function of the Second Kind”, American Statistical Association, Proceedings of the Statistical Computing Section, 89-94.

- Brill, P.H. (1990), “Examples of Level Crossing Estimation in M/G/1 Queues”, American Statistical Association, Proceedings of the Statistical Computing Section, 151-154.

- Brill, P.H. , Mandelbaum, M. (1987), “Measures of Flexibility in Production Systems”, Proceedings of the 9th International Conference on Production Research, 2474-2481.

B. __TECHNICAL REPORTS__

- Hlynka, M., Brill, P.H. (2008). “A Result for a Counter Problem”, University of Windsor Mathematics and Statistics Report, WMSR-08-01.

- Hlynka, M., Brill, P.H. (2007). “Stability in M/M/1 Queues with Reneging”, University of Windsor Mathematics and Statistics Report, WMSR-07-09.

- Mandelbaum, M., Hlynka, M., Brill, P.H. (2006). “Nonhomogeneous geometric Distribution Representation and Queueing”, University of Windsor Mathematics and Statistics Report, WMSR-06-03.

- Wu, X., Brill, P.H., Hlynka, M. (2005). “An M/ DN /1 Retrial Queue”, University of Windsor Mathematics and Statistics Report, WMSR-05-05.

- Wu, X., Brill, P.H., Hlynka, M., Wang, J. (2004). “An M/G/1 Retrial Queue with Retrials of the Customer and Balking”, Univ. of Windsor Mathematics and Statistics Report, WMSR-04-03.

- Brill, P.H. (2002) “Properties of the Waiting Time in M/DN/1 Queues”, University of Windsor Mathematics and Statistics Report, WMSR 02-01.

- Huang, M.L., Brill, P.H. (1999c). “A Level Crossing Method”, University of Windsor Mathematics and Statistics Report WMSR 99-01.

- Brill, P.H., Hlynka, M. (1998). “Companion Results to L = λ W”, University of Windsor Mathematics and Statistics Report WMSR 98-01.

- Brill, P.H., Hlynka, M. (1997). “A Multiple Server Who’s Next Queue”, University of Windsor Mathematics and Statistics Report WMSR 97-03.

- Brill, P.H., Hlynka, M. (1997). “A Geometric Interpretation of Markov Transition Matrices”, University of Windsor Mathematics and Statistics Report WMSR 97-01.

- Chen, Pedie, Brill, P.H. (1997). “Sieve of Eratosthenes and Bose’s Packing Problem” Colorado State University, Department of Statistics, Technical Report, #97-12.

- Huang, M. L., Brill, P. H. (1999). “A Level Crossing Method”, Windsor Mathematics & Statistics Report, University of Windsor, No. WMSR#99-01, pp. 1-28, 1999.

- Huang, M.L., Brill, P.H. (1996). “A Density Estimator Based on Level Crossings”, University of Windsor, School of business, Working Paper #W96-02, ISSN #1714-6191, (35 pp).

- Huang, M.L., Brill, P.H. (1996). “Empirical Distribution Based on Level Crossings”, University of Windsor, School of Business, Working Paper #W96-01, ISSN #1714-6191, (19 pp).

- Brill, P.H., Harris, C.M. (1995), “Queues with Markov-Connected Server Vacations: Theory and Computations”, Technical Report TR #089501,2, Department Of Operations Research and Engineering, George Mason University, Fairfax, Virginia, (23 pp).

- Chaouch, B.A., Brill, P.H. (1994), “Stationary Properties of an Inventory System with Irregular Shipment Patterns”, Univ. of Windsor, Sch. of Business, WP #94-13, ISSN #0714-6101, (21 pp).

- Brill, P.H., Hlynka, M. (1993), “The Who’s Next Queue with Two Lines”, Univ. of Windsor, School of Business, WP #93-08, ISSN #1714-6191, (20 pp).

- Brill, P.H., Chaouch, B.A. (1993), “An EOQ Model with Random Variations in Demand”, University of Windsor, School of business, WP #0714-6191, (30 pp).

- Brill, P.H., Harris, C.M. (1989), “Waiting Times for M/G/1 Queues with Service-Time-Dependent Server Vacations”, Technical Report #GMU/22474/110, Dept. of Op. Research & Applied Statistics, George Mason University, Fairfax, Virginia, USA, (18 pp). Accession Number: ADA209597.

- Brill, P.H. (1988a). “The Time Dependent System Point Method in Queues, Dams and Inventories”, Univ. of Windsor, School of Business, WP #88-16, (25 pp).

- Brill, P.H. (1988b), “The Time Dependent System Point Level Crossing Method for Exponential Queues”, University of Windsor, Scool of Business, WP #88-18, (33 pp).

- Brill, P.H. (1988c), “A Technique for Transient Distributions in Stochastic Models”, University of Windsor, School of Business, WP #88-13, (20 pp).

- Brill, P.H. (1988). "Waiting Times in Queues with State Dependent Bulk Service", University of Windsor, School of Business, WP #88-02, (23 pp).

- Mandelbaum, M., Brill, P.H. (1988). "Motivational Examples for Flexibility and Adaptivity in Production Systems." University of Windsor, School of Business, WP #88-01, (29 pp).

- Brill, P.H., Mandelbaum, M. (1987). "Adaptivity and Flexibility in Production Systems." Univ. of Windsor, School of Bus., WP#87-17 (18 pp).

- Brill, P.H. (1987). "Single Server Queues with Delay Dependent Arrival Streams", University of Windsor, School of Business, WP #87-16 (36 pp).

- Brill, P.H. (1987). "System-Point Computation of Virtual Waiting Time Distributions in Multiple Server Queues", University of Windsor, School of Business, WP #87-15 (32 pp).

- Brill, P.H. (1987). "System-Point Computation of Distributions in Queues, Dams and Inventories." University of Windsor, Odette School of Business, WP #87-12 (32 pp).

- Brill, P.H. (1987). "Bounds for Distributions of Waiting Times in Priority Queues", University of Windsor, School of Business, WP #87-11 (31 pp).

- Brill, P.H., Bachan, M. (1987). "Waiting Time Distributions in M/M/2 Priority Queues: A Level Crossing Approach." University of Windsor, School of Business, WP #87-09 (18 pp).

- Brill, P.H., Covert, D. (1987). "A Level Crossing Modelling Technique for Waiting Times in Two Class M/G/1 Priority Queues." Univ. of Windsor, School of Business, WP #87-10 (28 pp).

- Brill, P.H., Mandelbaum, M. (1987). "On Measures of Flexibility in Manufacturing Systems." Univ. of Windsor, School of Business, WP #87-08 (18 pp).

- Brill, P.H., Mandelbaum, M. (1987). "Measures of Flexibility in Production Systems." Univ. of Windsor, School of Business, WP #87-01 (8 pp).

- Brill, P.H. (1983). "System-Point Monte Carlo Simulation of Stationary Distributions of Waiting Times in Single Server Queues." University of Waterloo, Department of Statistics and Actuarial Science, Technical Report #83-10 (30 pp).
- Azoury, K., Brill, P.H. (1983). "A System-Point Approach to Inventory Models", University of Waterloo, Department of Statistics and Actuarial Science, Technical Report #83-07 (47 pp).

- Brill, P.H. (1983). "Queues with Reneging Depending on Required Wait", University of Waterloo, Department of Statistics and Actuarial Science, Technical Report #83-06, (72 pp).

- Green, L., Brill, P.H., (1982). "Queues in which Customers Receive Simultaneous Service from a Random Number of Servers - A System Point Approach." Columbia University, Graduate School of Business, Research Working Paper #466A (25 pp).

- Brill, P.H. (1976). "Embedded Level Crossing Processes in Dams and Queues", University of Toronto, Department of Industrial Engineering, WP #76-022 (24 pp).

- Brill, P.H., Posner, M.J.M. (1976). "A Characterization of Exponential Queues with Heterogeneous Servers." Univ. of Toronto, Dept. of Industrial Engineering, WP #76-021 (24 pp).

- *Brill, P.H., Posner, M.J.M. (1975). "Level Crossing in Point Process Applied to Queues: Single Server Case." University of Toronto, Department of Industrial Engineering, WP #75-009 (25 pp).

- *Brill, P.H., Posner, M.J.M. (1974). "On the Equilibrium Waiting Time Distribution for a Class of Exponential Queues." Univ. of Toronto, Dept. of Industrial Engineering, WP #74-012 (34 pp). (* See NOTES below.)

- *Brill, P.H., Posner, M.J.M. (1974). "Applications of a Conjecture on the Waiting Time Probability Density Function for Queues in Equilibrium." University of Toronto, Dept. of Industrial Engineering, WP #74-011 (29 pp). (* See NOTES below.)

- ** Brill, P.H., Posner (1974). "A Multiple Server Queue with Service Time Depending on Waiting Time" University of Toronto, Department of Industrial Engineering, WP #74-008 (30 pp). (** See NOTES below.)

- ** Brill, P.H., Posner, M.J.M. (1974). "Two Server Queues with Service Time Depending on Waiting Time", University of Toronto, Dept. of Industrial Engineering, WP #74-005 (61 pp). (** See NOTES below.)

C. __NOTES REGAREDING ORIGIN OF THE LEVEL CROSSING METHOD__

- *Working papers #74-011, #74-012 and #75-009, University of Toronto, introduce the concept relating sample path down- and up-crossing rates to stationary probability distributions of state variables in stochastic processes with continuous state spaces. Working paper #74-011 was the forerunner of the basic level crossing theorem, which appears first in WP #74-012 and in P. H. Brill’s PhD thesis (May, 1975). Working paper #74-012 demonstrates for the first time, the new method for obtaining stationary probability distributions of waiting times in simple and complex queues using a level crossing approach.

- **Working papers #74-005 and 74-008 precede the discovery of the level crossing methodology. The method of analysis in these two working papers was the classical embedded Markov chain technique based on embedded Markov chains and Lindley recursions, by means of which my original PhD thesis was all but completed. After discovering the level crossing method the entire PhD thesis was rewritten between November 1974 and March 1975, and titled “System Point Theory in Exponential Queues”. (The System Point is the leading point of a sample path, thought of evolutionarily over time.)

D. __BOOKS PUBLISHED: RESEARCH MONOGRAPHS__

1. Brill, P. H. (2008). Level Crossing Methods in Stochastic Models, International Series in Operations Research and Management Science ISOR 123, Springer, New York.

ISBN-978-0-387-09421-2 (477 pages, 106 figures).

2. Brill, P. H. (2017). Level Crossing Methods in Stochastic Models, Second Edition, International Series in Operations Research and Management Science ISOR 250, Springer, New York.

ISBN 978-3-319-50332-5 (559 pages, 124 figures, 27 in color).

E. __THESES__

__PHD THESIS__

Percy H. Brill (1975). System Point Theory in Exponential Queues, Department of Industrial Engineering, University of Toronto (viii +234 pages).

This thesis introduces the system-point level-crossing method for obtaining stationary probability distributions of state variables in stochastic models. It introduces the basic level crossing limit theorem, which proves the equality of the rate of sample-path downcrossings of a state-space level, to simple functions of the probability density function (pdf) of the state variable(s) at that level. It also introduces a decomposition-synthesis method for the state space (now called the method of pages or sheets), in models with complex state spaces. The subsequent solution technique for obtaining the pdfs of the state variables, applies sample-path rate balance across state-space levels, or sample-path entrance/exit rate balance of state-space measurable sets, to derive Volterra integral equations of the second kind for the pdfs of the state variable(s), by inspection of the sample path! This rate-balance technique applies to processes with piecewise continuous sample paths. It generalizes the rate-balance technique used in discrete-state continuous-time models, such as continuous-time Markov chains. The theory developed in this thesis connects particular (physical) sample-path motions, to each algebraic term of the Volterra integral equations for the pdf(s) of the state variable(s). Thus it gives concrete meaning to each algebraic term, with respect to a (physical) typical sample path of the process. Rate balance together with the one-to-one correspondence between sample path motions and algebraic terms in the equations, make the method intuitive, straightforward, and efficient for deriving the integral equations for the pdfs.

__MASTER’S THESIS__

Percy H. Brill (1959). Estimation of Proportions in Mixtures with Two Components, Department of Mathematical Statistics, in Columbia University Library (iii+59 pages).

ABSTRACT: The probability distribution of each component of the mixture is assumed to be known. Random numerical observations are taken from the mixture, but the observer is unable to distinguish to which component each observation belongs. The problem is to estimate the proportion of each component in the mixture, using the sample of observations as data. An example of the velocity distribution of a mixture of a gas, whose components are monomers and dimers, is given (in thermodynamics). Two methods of estimation of the proportion of each component are given. Graphs and tables are computed using the FORTRAN scientific computer language.

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