Fluid models of many-server queues with abandonment

Zhang, Jiheng

doi:10.1007/s11134-012-9307-9

“…These tools embrace and incorporate various areas in probability and OR such as inventory management, chain supply, queueing theory, level crossing theory, perishable commodities and optimization of cost functions which take into account all related costs (c.f. (Bar-Lev and Perry, 1993;Bassok et al, 1999;Berk and Gürler, 2008;Claeys et al, 2010;Cohen, 1977;Doshi, 1992;Kaspi and Perry, 1983;1984;Nahmias, 2011;Nahmias et al, 2004a;Parlar et al, 2010;Perry, 1985;1997;Perry and Posner, 1990;Perry and Stadje, 1999;2000a;2000b;Perry et al, 2000;Perry and Stadje, 2001;Zhang, 2013). The next subsections describe some features and ingredients of the stochastic approach.…”

Section: A Stochastic Approach To Blood Screening and Inventorymentioning

confidence: 99%

“…(Cohen, 1977;Doshi, 1992), under the assumption that the item and demand arrival processes are Poisson processes and that the shelf life time of items is constant (deterministic). Several extensions and ramifications have been studied; see for example (Bar-Lev and Perry, 1989;Berk and Gürler, 2008;Deniz et al, 2010;Doshi, 1992;Kaspi and Perry, 1983;1984;Nahmias, 2011;Nahmias et al, 2004;Parlar et al, 2010;Perry, 1985;1997;Perry 1999;Perry and Posner, 1990;Perry and Stadje, 1999;2000a;2000b;Perry et al, 2000;Perry and Stadje, 2001;Perry, 2003;Boxma et al, 2001;Zhang, 2013). The main mathematical tools employed in those studies are level crossing theory for Markov processes, martingales and stopping times.…”

Section: Stochastic Input-output Of the Inventory Systemmentioning

confidence: 99%

“…Their engagement has resulted in several papers, (c.f., (Bar-Lev et al, 2009;2017;2017a;2017b;Bar-Lev and Perry, 1989;Bar-Lev et al, 2005;2007;2013;. Other works can be found in (Abolnikov and Dukhovny, 2003;Kopach et al, 2008;Beliën and Forcé, 2012;Civelek et al, 2015;Ghandforoush and Sen, 2010;Karaesmen et al, 2011;Marshall et al, 2004;Sebastian et al, 2012;Shi and Zhao, 2010;Deniz et al, 2010;Blake and Hardy, 2014;Xie et al, 2012).…”

Section: Some Research Work Related To the Stochastic Approachmentioning

confidence: 99%

See 1 more Smart Citation

A Stochastic Approach to Blood Supply, Demand and Screening in Central Blood Banks - A Review

Bar‐Lev¹

2017

Current Research in Biostatistics

1

0

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Abstract:One of the major issues in securing blood supply to patients worldwide is to provide blood of the best achievable quality, in the needed quantities. Central Blood Services (CBS's) worldwide are daily faced with the problem how to satisfy demands for blood from various hospitals. These hospitals, in their turn, are faced with the problem how to satisfy demands for blood from their patients. To solve these problems in a costeffective way is notoriously difficult, because (i) the amounts of available blood and of blood demand are random, (ii) blood can only be used during a limited amount of time, (iii) one must distinguish various blood components (red blood cells, plasma and platelets) with different associated costs and perishability and (iv) one must distinguish persons with different blood types (like AB + and O − ) with different capabilities to act as donor or as recipient. In this review paper we provide the subject background, describing the blood characteristics and the operation of CBS and hospital blood banks. In particular we describe blood demand, blood components and blood types. We depict blood screening procedures and their processing times and provide with some real data. Particularly we describe a stochastic approach to blood screening and inventory. An emphasis will be given to inventory management and blood allocation, stochastic imput-output of the inventory system and some cost functions involved.

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“…Given the assumptions on the service times, the stochastic process [30,31,38]. It is an alternative to measure-valued approaches in [3,17,18,43] and distribution-valued approach in [35]. The alternative approaches are appealing for simplifying arguments and revealing structure; e.g., [35] shows that the the heavy-traffic limit for the G/G/∞ model can be regarded as a tempered-distribution-valued Ornstein-Uhlenbeck diffusion process, generalizing the diffusion process limit for the M/GI/∞ model in [3].…”

Section: Introductionmentioning

confidence: 99%

Heavy-traffic Limit for the Initial Content Process

Aras

¹

,

Liu

²

,

Whitt

³

2017

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To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a nonexponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space D D , assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed. (FCLT's) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables. From [14,41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Introduction. Heavy-traffic (HT) functional central limit theorems

show abstract

“…Given the assumptions on the service times, the stochastic process (X [30,31,38]. It is an alternative to measure-valued approaches in [3,17,18,43] and distribution-valued approach in [35]. The alternative approaches are appealing for simplifying arguments and revealing structure; e.g., [35] shows that the the heavy-traffic limit for the G/G/∞ model can be regarded as a tempered-distribution-valued Ornstein-Uhlenbeck diffusion process, generalizing the diffusion process limit for the M/GI/∞ model in [3].…”

mentioning

confidence: 99%

Heavy-traffic limit for the initial content process

Aras¹,

Liu²,

Whitt³

2017

Stoch. Syst.

0

View full text Add to dashboard Cite

To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a nonexponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space D D , assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed. (FCLT's) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables. From [14,41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Introduction. Heavy-traffic (HT) functional central limit theorems

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Fluid models of many-server queues with abandonment

Cited by 50 publications

References 30 publications

A Stochastic Approach to Blood Supply, Demand and Screening in Central Blood Banks - A Review

A Stochastic Approach to Blood Supply, Demand and Screening in Central Blood Banks - A Review

Heavy-traffic Limit for the Initial Content Process

Heavy-traffic limit for the initial content process

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