Exponential Single Server Queues in an Interactive Random Environment

References

• Anderson WJ (1991) Continuous-Time Markov Chains - An Application-Oriented Approach, vol. 7 of Springer Series in Statistics - Probability and its Applications (Springer, New York).Google Scholar
• Asmussen S (2003) Applied Probability and Queues, vol. 51 of Applications of Mathematics, 2nd ed. (Springer, New York).Google Scholar
• Baskett F, Chandy M, Muntz R, Palacios FG (1975) Open, closed and mixed networks of queues with different classes of customers. J. ACM. 22(2):248–260.Google Scholar
• Berman O, Kim E (1999) Stochastic models for inventory management at service facilities. Commun. Stat. Stoch. Models. 15(4):695–718.Google Scholar
• Berman O, Sapna KP (2000) Inventory management at service facilities for systems with arbitrarily distributed service times. Comm. Statist. Stoch. Models 16(3–4):343–360.Google Scholar
• Berman O, Sapna KP (2002) Optimal service rates of a service facility with perishable inventory items. Naval Res. Logist. 49(5):464–482.Google Scholar
• Boon M, Boxma OJ, Winands EMM (2011) On open problems in polling systems. Queueing Systems 68(3–4):365–374.Google Scholar
• Chung KL (1967) Markov Chains with Stationary Transition Probabilities (Springer, Berlin).Google Scholar
• Cogburn R (1980) Markov chains in random environments: The case of Markovian environments. Ann. Probab. 8(5):908–916.Google Scholar
• Cogburn R (1984) The ergodic theory of Markov chains in random environments. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 66(1):109–128.Google Scholar
• Cogburn R, Torrez WC (1981) Birth and death processes with random environments in continuous time. J. Appl. Probab. 18(1):19–30.Google Scholar
• Cornez R (1987) Birth and death processes in random environments with feedback. J. Appl. Probab. 24(1):25–34.Google Scholar
• Daduna H (2016) Moving queue on a network. Remke A, Haverkort BR, eds. Measurement, Modelling and Evaluation of Dependable Computer and Communication Systems, vol. 9629 of Lecture Notes in Computer Science (Springer, Cham, Switzerland), 40–54.Google Scholar
• Di Crescenzo A, Iuliano A, Martinucci B (2012) On a bilateral birth-death process with alternating rates. Ric. Mat. 61(1):157–169.Google Scholar
• Di Crescenzo A, Macci C, Martinucci B (2014) Asymptotic results for random walks in continuous time with alternating rate. J. Statist. Phys. 154(5):1352–1364.Google Scholar
• Doshi BT (1990) Single server queues with vacations. Takagi H, ed. Stochastic Analysis of Computer and Communication Systems (North–Holland, Amsterdam), 217–267.Google Scholar
• Economou A (2003) A characterization of product-form stationary distributions for queueing systems in random environment. Proc. 17th Eur. Simulation Multiconf. (SCS-European Publishing House, Delft, Netherlands), 193–198.Google Scholar
• Economou A (2005) Generalized product-form stationary distributions for Markov chains in random environments with queueing applications. Adv. Appl. Probab. 37(1):185–211.Google Scholar
• Falin G (1996) A heterogeneous blocking system in a random environment. J. Appl. Probab. 33(1):211–216.Google Scholar
• Foss S, Shneer S, Tyurlikov A (2012) Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols. Stoch. Syst. 2(1):208–231.Link, Google Scholar
• Gannon M, Pechersky E, Suhov Y, Yambartsev V (2016) A random walk in a queueing network environment. J. Appl. Probab. 53(2):448–462.Google Scholar
• Gaver DP, Jacobs RA, Latouche G (1984) Finite birth-and-death models in randomly changing environments. Adv. Appl. Probab. 16(4):715–731.Google Scholar
• Helm W, Waldmann KH (1984) Optimal control of arrivals to multiserver queues in a random environment. J. Appl. Probab. 21(3):602–615.Google Scholar
• Jackson JR (1957) Networks of waiting lines. Oper. Res. 5(4):518–521.Link, Google Scholar
• Jeganathan K (2014) Perishable inventory system at service facilities with multiple server vacations and impatient customers. J. Statist. Appl. Probab. Lett. 3(3):63–73.Google Scholar
• Kelly FP (1976) Networks of queues. Adv. Appl. Probab. 8(2):416–432.Google Scholar
• Kelly FP (1979) Reversibility and Stochastic Networks (Wiley, Chichester, UK).Google Scholar
• Kelly FP, Yudovina E (2014) Stochastic Networks. IMS Textbooks (Cambridge University Press, Cambridge, UK).Google Scholar
• Koroliuk VS, Melikov AZ, Ponomarenko LA, Rustamov AM (2017) Asymptotic analysis of the system with server vacation and perishable inventory. Cybernet. Systems Anal. 53(4):543–553.Google Scholar
• Koroliuk VS, Melikov AZ, Ponomarenko LA, Rustamov AM (2018) Models of perishable queueing-inventory systems with server vacation. Cybernet. Systems Anal. 54(1):31–44.Google Scholar
• Krenzler R (2016) Queueing systems in a random environment. PhD thesis, Universität Hamburg, Department of Mathematics, Hamburg, Germany.Google Scholar
• Krenzler R, Daduna H (2012) Loss Systems in a Random Environment—Steady State Analysis (Center of Mathematical Statistics and Stochastic Processes, Department of Mathematics, Hamburg University, Hamburg).Google Scholar
• Krenzler R, Daduna H (2014) Modeling and performance analysis of a node in fault tolerant wireless sensor networks. Fischbach K, Krieger UR, eds. Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault-Tolerance (Springer, Berlin), 73–87.Google Scholar
• Krenzler R, Daduna H (2015a) Loss systems in a random environment - steady state analysis. Queueing Systems 80(1–2):127–153.Google Scholar
• Krenzler R, Daduna H (2015b) Performability analysis of an unreliable M/M/1-type queue. Wolfinger BE, Heidtmann K-D, eds. Leistungs-, Zuverlässigkeits- und Verlässlichkeitsbewertung von Kommunikationsnetzen und verteilten Systemen, vol. 302. (Berichte des FB Informatik der Universität Hamburg, Hamburg, Germany), 90–95.Google Scholar
• Krishnamoorthy A, Lakshmy B, Manikandan R (2011) A survey on inventory models with positive service time. Opsearch 48(2):153–169.Google Scholar
• Krishnamoorthy A, Pramod PK, Chakravarthy SR (2014) Queues with interruptions: A survey. TOP 22(1):290–320.Google Scholar
• Krishnamoorthy A, Shajin D, Viswanath CN (2019) Inventory with positive service time: a survey. Anisimov V, Limnios N, eds. Advanced Trends in Queueing Theory: Series of Books “Mathematics and Statistics”, chapter 5 (ISTE & Wiley, London), 171–208.Google Scholar
• Manuel P, Sivakumar B, Arivarignan G (2007) A perishable inventory system with service facilities, MAP arrivals and PH-service times. J. Syst. Sci. Syst. Engrg. 16(1):62–73.Google Scholar
• Manuel P, Sivakumar B, Arivarignan G (2008) A perishable inventory system with service facilities and retrial customers. Comput. Ind. Engrg. 54(3):484–501.Google Scholar
• Melikov AZ, Molchanov AA (1992) Stock optimization in transportation/storage systems. Cybernet. Systems Anal. 28(3):484–487.Google Scholar
• Neuts MF (1981) Matrix Geometric Solutions in Stochastic Models - An Algorithmic Approach (Johns Hopkins University Press, Baltimore).Google Scholar
• Neuts MF, Lucantoni DM (1979) Markovian queue with n servers subject to breakdowns and repairs. Management Sci. 25(9):849–861.Link, Google Scholar
• Neuts MF, Rao BM (1990) Numerical investigation of a multiserver retrial model. Queueing Systems 7(2):169–189.Google Scholar
• Otten S (2018) Integrated models for performance analysis and optimization of queueing-inventory-systems in logistic networks. PhD thesis, Universität Hamburg, Department of Mathematics, Hamburg, Germany.Google Scholar
• Pang G, Sarantsev A, Belopolskaya Y, Suhov Y (2020) Stationary distributions and convergence for M/M/1 queues in interactive random environments. Queueing Systems 94(3–4):357–392.Google Scholar
• Prabhu NU, Zhu Y (1989) Markov-modulated queueing systems. Queueing Systems 5(1):215–245.Google Scholar
• Prabhu NU, Zhu Y (1995) Corrections to our paper: Markov-modulated queueing systems. Queueing Systems 19(4):449.Google Scholar
• Saffari M, Asmussen S, Haji R (2013) The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Systems 75(1):65–77.Google Scholar
• Saffari M, Haji R, Hassanzadeh F (2011) A queueing system with inventory and mixed exponentially distributed lead times. Internat. J. Adv. Manufacturing Tech. 53(9–12):1231–1237.Google Scholar
• Sauer C, Daduna H (2003) Availability formulas and performance measures for separable degradable networks. Econom. Quality Control 18(2):165–194.Google Scholar
• Schwarz M, Sauer C, Daduna H, Kulik R, Szekli R (2006) M/M/1 Queueing systems with inventory. Queueing Systems 54(1):55–78.Google Scholar
• Sigman K, Simchi-Levi D (1992) Light traffic heuristic for an M/G/1 queue with limited inventory. Ann. Oper. Res. 40(1):371–380.Google Scholar
• Spieksma FM, Tweedie RL (1994) Strengthening ergodicity to geometric ergodicity for Markov chains. Comm. Statist. Stoch. Models 10(1):45–74.Google Scholar
• Sznitman A-S (2002) Lectures on random motions in random media. Sznitman A-S, Bolthausen E, eds. Ten Lectures on Random Media, vol. 32 of DMV-Seminar (Birkhäuser Verlag, Basel), 9–51.Google Scholar
• Takagi H (1990) Queueing analysis of polling models: An update. Takagi H, ed. Stochastic Analysis of Computer and Communications Systems (North-Holland, Amsterdam), 267–318.Google Scholar
• Vineetha K (2008) Analysis of inventory systems with positive and negligible service time. PhD thesis, Department of Statistics, University of Calicut, Tenhipalam, India.Google Scholar
• van der Wal J (1989) Monotonicity of the throughput of a closed exponential queueing network in the number of jobs. OR Spectrum 11(6):97–100.Google Scholar
• van Dijk NM (1993) Queueing Networks and Product Forms – A Systems Approach (Wiley, Chichester, UK).Google Scholar
• van Dijk NM (1998) Bounds and error bounds for queueing networks. Ann. Oper. Res. 79(0):295–319.Google Scholar
• van Dijk NM (2011) On practical product form characterizations. Boucherie RJ, van Dijk NM, eds. Queueing Networks: A Fundamental Approach, vol. 154 of International Series in Operations Research and Management Science (Springer, New York), 1–83.Google Scholar
• van Dijk NM, Korezlioglu H (1992) On product form approximations for communication networks with losses: error bounds. Ann. Oper. Res. 35(1):69–94.Google Scholar
• van Dijk NM, van der Wal J (1989) Simple bounds and monotonicity results for multi-server exponential tandem queues. Queueing Systems 4(1):1–16.Google Scholar
• Yadavalli VSS, Anbazhagan N, Jeganathan K (2015) A two heterogeneous servers perishable inventory system of a finite population with one unreliable server and repeated attempts. Pakistan J. Statist. 31(1):135–158.Google Scholar
• Yadavalli VSS, Sivakumar B, Arivarignan G (2007) Stochastic inventory management at a service facility with a set of reorder levels. ORiON 28(2):137–149.Google Scholar
• Yadavalli VSS, Sivakumar B, Arivarignan G, Adetunji O (2012) A finite source multi-server inventory system with service facility. Comput. Ind. Engrg. 63(4):739–753.Google Scholar
• Yechiali U (1973) A queuing-type birth-and-death process defined on a continuous-time Markov chain. Oper. Res. 21(2):604–609.Link, Google Scholar
• Zhu Y (1994) Markovian queueing networks in a random environment. Oper. Res. Lett. 15(1):11–17.Google Scholar