×

A blood bank model with perishable blood and demand impatience. (English) Zbl 1390.60340

Summary: We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to independent compound Poisson processes. Blood is perishable, i.e., blood can only be kept in storage for a limited amount of time. Furthermore, demand for blood is impatient, i.e., a demand for blood may be canceled if it cannot be satisfied soon enough. For a range of perishability functions and demand impatience functions, we derive the steady-state distributions of the amount of blood kept in storage, and of the amount of demand for blood (at any point in time, at most one of these quantities is positive). Under certain conditions we also obtain the fluid and diffusion limits of the blood inventory process, showing in particular that the diffusion limit process is an Ornstein-Uhlenbeck process.

MSC:

60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60J60 Diffusion processes
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)

Software:

DLMF
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] Albrecher, H., Lautscham, V. (2013) From ruin to bankruptcy for compound Poisson surplus processes.ASTIN Bull.43:213-243. · Zbl 1283.91084
[2] Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H., de Turck, K. (2016) A functional central limit theorem for a Markov-modulated infinite-server queue.Methodology Comput. Appl. Probab.18(1):151-168. · Zbl 1336.60066
[3] Asmussen, S. (2003)Applied Probability and Queues, 2nd ed. (Springer, New York). · Zbl 1029.60001
[4] Bar-Lev, SK., Boxma, OJ., Mathijsen, BWJ., Perry, D. (2015) A blood bank model with perishable blood and demand impatience. Eurandom Report 2015-015. · Zbl 1390.60340
[5] Bekker, R., Borst, SC., Boxma, OJ., Kella, O. (2004) Queues with workload-dependent arrival and service rates.Queueing Systems46:537-556. · Zbl 1056.90026
[6] Beliën, J., Forcé, H. (2012) Supply chain management of blood products: A literature review.Eur. J. Oper. Res.217:1-16.
[7] Billingsley, P. (1999)Convergence of Probability Measures, Wiley Series in Probability and Statistics, 2nd ed. (John Wiley &Sons, New York). · Zbl 0944.60003
[8] Boucherie, RJ., Boxma, OJ. (1996) The workload in theM/G/1 queue with work removal.Prob. Engr. Inf. Sci.10:261-277. · Zbl 1095.60510
[9] Boxma, OJ., Essifi, R., Janssen, AJEM. (2016) A queueing/inventory and an insurance risk model.Adv. Appl. Probab.48(4):1139-1160. · Zbl 1358.60095
[10] Boxma, OJ., David, I., Perry, D., Stadje, W. (2011) A new look at organ transplant models and double matching queues.Probab. Engrg. Informational Sci.25:135-155. · Zbl 1213.90085
[11] Chen, H., Yao, DD. (2001) Fundamentals of queueing networks: Performance, asymptotics and optimization.Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization.Stochastic Modelling and Applied Probability, Vol. 46 (Springer, New York). · Zbl 0992.60003
[12] Dai, JG., Dieker, AB., Gao, X. (2014) Validity of heavy-traffic steady-state approximations in many-server queues with abandonment.Queueing Systems78:1-29. · Zbl 1310.60127
[13] Dai, JG., He, S., Tezcan, T. (2010) Many-server diffusion limits forG/Ph/n+GIqueues.Ann. Appl. Probab.20:1854-1890. · Zbl 1202.90085
[14] Gamarnik, D., Goldberg, DA. (2013) Steady-stateGI/G/Nqueue in the Halfin-Whitt regime.Ann. Appl. Probab.23:2382-2419. · Zbl 1285.60090
[15] Gamarnik, D., Zeevi, A. (2006) Validity of heavy traffic steady-state approximations in generalized Jackson networks.Ann. Appl. Probab.16:56-90. · Zbl 1094.60052
[16] Garnett, O., Mandelbaum, A., Reiman, MI. (2002) Designing a call center with impatient customers.Manufacturing Service Oper. Management4(3):208-227.
[17] Ghandforoush, P., Sen, TK. (2010) A DSS to manage platelet production supply chain for regional blood centers.Decision Support Systems50:32-42.
[18] Gurvich, I. (2014) Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines.Math. Oper. Res.39(1):121-162. · Zbl 1312.60111
[19] Keilson, J., Mermin, ND. (1959) The second-order distribution of integrand shot noise.IRE Trans.IT- 5:75-77.
[20] Olver, FW., Lozier, DW., Boisvert, RF., Clark, CW. (2010)NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University Press, New York).
[21] Pang, G., Talreja, R., Whitt, W. (2007) Martingale proofs of many-server heavy-traffic limits for Markovian queues.Probab. Surveys4:193-267. · Zbl 1189.60067
[22] Puhalskii, AA., Reiman, MI. (2000) The multiclassGI/PH/Nqueue in the Halfin-Whitt regime.Adv. Appl. Probab.32:564-595. · Zbl 0962.60089
[23] Reed, J., Zwart, AP. (2011) A piecewise linear stochastic differential equation driven by a Lévy process.J. Appl. Probab.48A:109-119. · Zbl 1229.60075
[24] Ross, SM. (1996)Stochastic Processes, Wiley Series in Probability and Mathematical Statistics, 2nd ed. (John Wiley & Sons, New York).
[25] Slater, LJ. (1960)Confluent Hypergeometric Functions(Cambridge University Press, Cambridge, UK). · Zbl 0086.27502
[26] Stanger, SHW., Yates, N., Wilding, R., Cotton, S. (2012) Blood inventory management: Hospital best practice.Transfusion Medicine Rev.26:153-163.
[27] Steiner, ME., Assmann, SF., Levy, JH., Marshall, J., Pulkrabek, S., Sloan, SR., Triulzi, D., Stowell, CP. (2010) Addressing the question of the effect of RBC storage on clinical outcomes: The red cell storage duration study (RECESS, Section 7).Transfus Apher Sci.43:107-116.
[28] Ward, AR., Glynn, PW. (2003) A diffusion approximation for a Markovian queue with reneging.Queueing Systems43:103-128. · Zbl 1054.60100
[29] Ward, AR., Glynn, PW. (2005) A diffusion approximation for aGI/GI/1 queue with balking or reneging.Queueing Systems50:371-400. · Zbl 1094.60064
[30] Whitt, W. (2002)Stochastic-Process Limits, Springer Series in Operations Research (Springer, New York). · Zbl 0993.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.