Home | Articles | Past volumes | About | Login | Notify | Contact | Search
 Stochastic Systems open journal systems 


A blood bank model with perishable blood and demand impatience

Shaul K. Bar-Lev, University of Haifa
Onno Boxma, Eindhoven University of Technology
Britt Mathijsen, Eindhoven University of Technology
David Perry, University of Haifa


Abstract
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 \(X_b\) kept in storage, and of the amount of demand for blood \(X_d\) (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.

AMS 2000 subject classifications: Primary 60K30; secondary 60J60, 33C15.

Keywords: Blood bank, level-crossings, shot-noise model, confluent hypergeometric functions, scaling limits, Ornstein-Uhlenbeck process.

Creative Common LOGO

Full Text: PDF


Bar-Lev, Shaul K., Boxma, Onno, Mathijsen, Britt, Perry, David, A blood bank model with perishable blood and demand impatience, Stochastic Systems, 7, (2017), 1-46 (electronic). DOI: 10.1214/15-SSY197.

References

[1]    H. Albrecher and V. Lautscham (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bulletin 43, 213–243. MR3388122

[2]    D. Anderson, J. Blom, M. Mandjes, H. Thorsdottir and K. de Turck (2016). A Functional Central Limit Theorem for a Markov-modulated infinite-server queue. Methodology and Computing in Applied Probability 18(1), 151–168. MR3465473

[3]    S. Asmussen (2003). Applied Probability and Queues. Springer, New York (2nd ed.). MR1978607

[4]    S. K. Bar-Lev, O. J. Boxma, B. W. J. Mathijsen and D. Perry (2015). A blood bank model with perishable blood and demand impatience. Eurandom Report 2015-015.

[5]    R. Bekker, S. C. Borst, O. J. Boxma and O. Kella (2004). Queues with workload-dependent arrival and service rates, Queueing Systems 46, 537–556. MR2068140

[6]    J. Beliën and H. Forcé (2012). Supply chain management of blood products: A literature review. European Journal of Operational Research 217, 1–16. MR2851634

[7]    P. Billingsley (1999). Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics. MR1700749

[8]    R. J. Boucherie and O. J. Boxma (1996). The workload in the M∕G∕1 queue with work removal, Prob. Engr. Inf. Sci. 10, 261–277. MR1386640

[9]    O. J. Boxma, I. David, D. Perry and W. Stadje (2011). A new look at organ transplant models and double matching queues. Probability in the Engineering and Informational Sciences 25, 135–155. MR2786745

[10]    O. J. Boxma, R. Essifi and A. J. E. M. Janssen (2017). A queueing/inventory and an insurance risk model. To appear in Journal of Applied Probability. MR3595769

[11]    H. Chen and D. D. Yao (2001). Fundamentals of queueing networks: Performance, asymptotics and optimization, Springer Series: Stochastic Modelling and Applied Probability, 46. MR1835969

[12]    J. G. Dai, A. B. Dieker and X. Gao (2014). Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. Queueing Systems 78, 1–29. MR3238006

[13]    J. G. Dai, S. He and T. Tezcan (2010). Many-server diffusion limits for G∕Ph∕n+ GI queues. The Annals of Applied Probability 20, 1854–1890. MR2724423

[14]    D. Gamarnik and D. A. Goldberg (2013). Steady-state GI∕G∕N queue in the Halfin-Whitt regime. The Annals of Applied Probability 23, 2382–2419. MR3127939

[15]    D. Gamarnik and A. Zeevi (2006). Validity of heavy traffic steady-state approximations in generalized Jackson networks. The Annals of Applied Probability 16, 56–90. MR2209336

[16]    O. Garnett, A. Mandelbaum and M. I. Reiman. Designing a call center with impatient customers. Manufacturing & Service Operations Management 4, 208–227.

[17]    P. Ghandforoush and T. K. Sen (2010). A DSS to manage platelet production supply chain for regional blood centers. Decision Support Systems 50, 32–42.

[18]    I. Gurvich (2013) Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines. Mathematics of Operations Research 39, 121–162. MR3173006

[19]    J. Keilson and N. D. Mermin (1959). The second-order distribution of integrand shot noise, IRE Trans. IT-5, 75–77. MR0124092

[20]    F. W. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (2010). NIST Handbook of Mathematical Functions. Cambridge University Press (1st ed.) MR2655349

[21]    G. Pang, R. Talreja and W. Whitt (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues, Probability Surveys 4, 193–267. MR2368951

[22]    A. A. Puhalskii and M. I. Reiman (2000). The multiclass GI/PH/N queue in the Halfin-Whitt regime. Advances in Applied Probability 32, 564–595. MR1778580

[23]    J. Reed and A. P. Zwart (2011). A piecewise linear stochastic differential equation driven by a Lévy process. Journal of Applied Probability 48A, 109–119. MR2865620

[24]    S. M. Ross (1996). Stochastic Processes. Wiley Series in Probability and Mathematical Statistics (2nd ed.), New York. MR1373653

[25]    L. J. Slater (1960). Confluent Hypergeometric Functions. Cambridge University Press, Cambridge. MR0107026

[26]    S. H. W. Stanger, N. Yates, R. Wilding and S. Cotton (2012). Blood inventory management: Hospital best practice. Transfusion Medicine Reviews 26, 153–163.

[27]    M. E. Steiner, S. F. Assmann, J. H. Levy, J. Marshall, S. Pulkrabek, S. R. Sloan, D. Triulzi and C. P. Stowell (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]    A. R. Ward and P. W. Glynn (2003). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43, 103–128. MR1957808

[29]    A. R. Ward and P. W. Glynn (2005). A diffusion approximation for a GIGI1 queue with balking or reneging. Queueing Systems 50, 371–400. MR2172907

[30]    W. Whitt (2002). Stochastic-Process Limits. Springer Series in Operations Research, New York. MR1876437




Home | Articles | Past volumes | About | Login | Notify | Contact | Search

Stochastic Systems. ISSN: 1946-5238