Loss systems in a random environment

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove an if-and-only-if-condition for a product form steady state distribution of the joint queueing-environment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, e.g. from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, e.g. of queueing-inventory processes. We investigate further classical loss systems, where due to finite waiting room loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise. We further investigate the embedded Markov chains at departure epochs and show that the behaviour of the embedded Markov chain is often considerably different from that of the continuous time Markov process. This is different from the behaviour of the standard M/G/1, where the steady state of the embedded Markov chain and the continuous time process coincide. For exponential queueing systems we show that there is a product form equilibrium of the embedded Markov chain under rather general conditions. For systems with non-exponential service times more restrictive constraints are needed, which we prove by a counter example where the environment represents an inventory attached to an M/D/1 queue. Such integrated queueing-inventory systems are dealt with in the literature previously, and are revisited here in detail

EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Applied Probability

[no abstract available

Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe

Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

An (S - 1; S) Inventory System with Negative Arrivals and Multiple Vacations

In this paper, we consider a continuous review one-to-one ordering policy inventory system with multiple vacations and negative customers. The maximum storage capacity is S. The customers arrive according to a Poisson process with finite waiting hall. There are two types of customers: ordinary and negative. An ordinary customer, on arrival, joins the queue and the negative customer does not join the queue and takes away any one of the waiting customers. When the waiting hall is full, the arriving primary customer is considered to be lost. The service time and lead time are assumed to have independent exponential distribution. When the inventory becomes empty, the server takes a vacation and the vacation duration is exponentially distributed. The stationary distribution of the number of customers in the waiting hall, the inventory level and the server status for the steady state case. Some system performance measures and numerical illustrations are discussed

Monotonicity and stability of periodic polling models

Polling Systems;Stability;operations research

Linear retrial inventory system with second optional service under mixed priority service

The present paper deals with a generalization of the homogeneous single server finite source retrial inventory system with two classes of customers - one with high priority customer and the other with low priority customer. The inventory is replenished according to an (s, Q) policy and the replenishing times are assumed to be exponentially distributed. The server provides two types of services - one with essential service and the other with a second optional service. The service times of the 1st (essential) and 2nd (optional) services are independent and exponentially distributed. The high priority customers have a mixed priority over the low priority customers. Retrial is introduced for low priority customers only. The joint probability distribution of the number of customers in the waiting hall, the number of customers in the orbit and the inventory level is obtained for the steady state case. Some important system performance measures in the steady state are derived and the long-run total expected cost rate is also derived.Publisher's Versio

Bloody fast blood collection

This thesis consists of four parts: The first part contains an introduction, the second presents approaches for the evaluation of waiting times at blood collection sites, the third uses these to present approaches that improve waiting times at blood collection sites. The final part shows the application of two of the approaches to data from real blood collection sites, followed by the conclusions that can be drawn from this thesis. Part I: Introduction, contains two chapters. Chapter 1 introduces the context for this thesis: blood banks in general, the Dutch blood bank Sanquin and blood collection sites. The chapter sketches some of the challenges faced with respect to blood collection sites. As blood donors are voluntary and non-remunerated, delays and waiting times within blood collection sites should be kept at acceptable levels. However, waiting times are currently not incorporated in staff planning or in other decisions with respect to blood collection sites. These blood collection sites will be the primary focus of this thesis. This thesis provides methods that do take waiting times into account, aiming to decrease waiting times at blood collection sites and leveling work pressure for staff members, without the need for additional staff. Chapter 2 then presents a technical methods that will be used most of the chapters in this thesis: uniformization. Uniformization can be used to transform Continuous Time Markov Chains (CTMCs) — that are very hard to analyze — into Discrete Time Markov Chains (DTMCs) — that are much easier to analyze. The chapter shows how the method works, provides an extensive overview of the literature related to the method, the (technical) intuition behind the method as well as several extensions and applications. Although not all of the extensions and applications are necessary for this thesis, it does provide an overview of one of the most valuable methods for this thesis. Part II: Evaluation, contains two chapters that propose and adapt several methods to compute waiting times and queues at blood collection sites. A blood collection site is best modeled as a time-dependent queueing network, requiring non-standard approaches. Chapter 3 considers a stationary, i.e. not time-dependent model of blood collection sites as a first step. A blood collection site consists of three main stations: Registration, Interview and Donation. All three of the stations can have their own queue. This means that even the stationary model is non-trivial for some computations. However, for the stationary model, an analytic so-called product form expression is derived. Based on this product form, two more results are shown. The first result is that the standard waiting time distributions from M|M|s queues are applicable, as if the queue is in isolation. It is then concluded that no closed form expression exist for the total waiting or delay time distribution, as the distributions of the three stations in tandem are not independent. Therefore a numerical approach is presented to compute the total delay time distribution of a collection site. All of the results are supported by numerical examples based on a Dutch blood collection site. The approach for the computation of the total delay time distribution can also be combined with the approach from Chapter 4 for an extension to a time-dependent setting. Chapter 4 shows an approach to deal with these time-dependent aspects in queueing systems, as often experienced by blood collection sites and other service systems, typically due to time-dependent arrivals and capacities. Easy and quick to use queueing expressions generally do not apply to time-dependent situations. A large number of computational papers has been written about queue length distributions for time-dependent queues, but these are mostly theoretical and based on single queues. This chapter aims to combine computational methods with more realistic time-dependent queueing networks, with an approach based on uniformization. Although uniformization is generally perceived to be too computationally prohibitive, we show that our method is very effective for practical instances, as shown with an example of a Dutch blood collection site. The objective of the results is twofold: to show that a time-dependent queueing network approach can be beneficial and to evaluate possible improvements for Dutch blood collection sites that can only be properly assessed with a time-dependent queueing method. Part III: Optimization, contains four chapters that aim to improve service levels at Sanquin. The first three chapters focus on three different methods to decrease queues at blood collection sites. Chapters 5 and 6 focus on improving the service by optimizing staff allocation to shifts and stations. Chapter 7 focuses on improving the arrival process with the same goal. Chapter 8 is focused at improving inventory management of red blood cells. Donors do not arrive to blood collection sites uniformly throughout the day, but show clear preferences for certain times of the day. However, the arrival patterns that are shown by historical data, are not used for scheduling staff members at blood collection sites. As a first significant step to shorten waiting times we can align staff capacity and shifts with walk-in arrivals. Chapter 5 aims to optimize shift scheduling for blood collection sites. The chapter proposes a two-step procedure. First, the arrival patterns and methods from queueing theory are used to determine the required number of staff members for every half hour. Second, an integer linear program is used to compute optimal shift lengths and starting times, based on the required number of staff members. The chapter is concluded with numerical experiments that show, depending on the scenario, a reduction of waiting times, a reduction of staff members or a combination of both. At a blood collection site three stations (Registration, Interview and Donation) can roughly be distinguished. Staff members at Dutch blood collection sites are often trained to work at any of these stations, but are usually allocated to one of the stations for large fractions of a shift. If staff members change their allocation this is based on an ad hoc decision. Chapter 6 aims to take advantage of this mostly unused allocation flexibility to reduce queues at blood collection sites. As a collection site is a highly stochastic process, both in arrivals and services, an optimal allocation of staff members to the three stations is unknown, constantly changing and a challenge to determine. Chapter 6 provides and applies a so-called Markov Decision Process (MDP) to compute optimal staff assignments. Extensive numerical and simulation experiments show the potential reductions of queues when the reallocation algorithm would be implemented. Based on Dutch blood collection sites, reductions of 40 to 80% on the number of waiting donors seem attainable, depending on the scenario. Chapter 7 also aims to align the arrival of donors with scheduled staff, similarly to Chapter 5. Chapter 7 tries to achieve this by changing the arrivals of donors. By introducing appointments for an additional part of donors, arrivals can be redirected from the busiest times of the day to quiet times. An extended numerical queueing model with priorities is introduced for blood collection sites, as Sanquin wants to incentive donors to make appointments by prioritizing donors with appointments over donors without appointments. Appointment slots are added if the average queue drops below certain limits. The correct values for these limits, i.e. the values that plan the correct number of appointments, are then determined by binary search. Numerical results show that the method succeeds in decreasing excessive queues. However, the proposed priorities might result in unacceptably high waiting times for donors without appointments, and caution is therefore required before implementation. Although this thesis mainly focuses on blood collection sites, many more logistical challenges are present at a blood bank. One of these challenges arises from the expectation that Sanquin can supply hospitals with extensively typed red blood cell units directly from stock. Chapter 8 deals with this challenge. Currently, all units are issued according to the first-in-first-out principle, irrespective of their specific typing. These kind of issuing policies lead to shortages for rare blood units. Shortages for rare units could be avoided by keeping them in stock for longer, but this could also lead to unnecessary wastage. Therefore, to avoid both wastage and shortages, a trade-off between the age and rarity of a specific unit in stock should be made. For this purpose, we modeled the allocation of the inventory as a circulation flow problem, in which decisions about which units to issue are based on both the age and rarity of the units in stock. We evaluated the model for several settings of the input parameters. It turns out that, especially if only a few donors are typed for some combinations of antigens, shortages can be avoided by saving rare blood products. Moreover, the average issuing age remains unchanged. Part IV: Practice and Outlook concludes this thesis. The first of two chapters in this part shows the combined application of two approaches from this thesis to data from three collection sites in the Netherlands. The final chapter of this thesis presents the conclusions that can be drawn from this thesis and discusses an outlook for further research. Chapter 9 shows the combined application of the methods in Chapters 5 and 6 to three real collection sites in Dutch cities: Nijmegen, Leiden and Almelo. The collection sites in Nijmegen and Leiden are both large fixed collection sites. The collection site in Almelo is a mobile collection site. The application of each one of the methods individually reduce waiting times significantly, and the combined application of the methods reduces waiting times even further. Simultaneously, small reductions in the number of staff hours are attainable. The results from Chapter 9 summarize the main message of this thesis: waiting time for blood donors at blood collection sites can be reduced without the need for more staff members when the working times of staff members are used more effectively and efficiently, and controlling the arrival process of donors. The approaches presented in this thesis can be used for this purpose. This is not only beneficial for blood donors, but will also result in more balanced workload for staff members, as fluctuations in this workload are reduced significantly