Location models for airline hubs behaving as M/D/c queues

Models are presented for the optimal location of hubs in airline networks, that take into consideration the congestion effects. Hubs, which are the most congested airports, are modeled as M/D/c queuing systems, that is, Poisson arrivals, deterministic service time, and {\em c} servers. A formula is derived for the probability of a number of customers in the system, which is later used to propose a probabilistic constraint. This constraint limits the probability of {\em b} airplanes in queue, to be lesser than a value $\alpha$. Due to the computational complexity of the formulation. The model is solved using a meta-heuristic based on tabu search. Computational experience is presented.Hub location, congestion, tabu-search

Analysis of a discrete-time single-server queue with an occasional extra server

We consider a discrete-time queueing system having two distinct servers: one server, the "regular" server, is permanently available, while the second server, referred to as the "extra" server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random "up-periods", whereas it is unavailable during random "down-periods". Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. For this queueing model, we are able to derive closed-form expressions for the steady-state probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials. Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples. Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modeling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty. (C) 2017 Elsevier B.V. All rights reserved

Queueing theory applied to pre-hospital and retrieval medicine

Background: Pre-hospital and Retrieval Medicine is a healthcare specialty focussed on the provision of advanced, specialist care to patients in any clinical setting - from the roadside to a major hospital. The ScotSTAR division of the Scottish Ambulance Service has a national remit for providing such services within Scotland. High clinical acuity and long travel time challenge the service by generating a significant workload from relatively few patients. ScotSTAR teams comprise only one or two servers, so waiting times are potentially long, and there is a relatively high per-patient cost. Aims: Firstly, this thesis aims to investigate if standard queueing theory could be used to describe two ScotSTAR teams: the Scottish Paediatric Retrieval Service (SPRS) and the Emergency Medical Retrieval Service (EMRS). The thesis then aims to develop a Discrete Event Simulation (DES) model and validate it against the real-world. From this model, the thesis then aims to describe the performance of the ScotSTAR teams using metrics which are unmeasurable in the real-world. Finally, the thesis aims to establish the performance frontiers of the ScotSTAR teams. Methods: Analysis of the ScotSTAR teams to map their operation with standard queueing theory was undertaken. This was used to develop a DES model, which performed 1000 simulation iterations of a 4-year period. The output was compared to the real-world data for accuracy with regard to: number of missions, activation time of day, inter-arrival time, mission duration, and server utilization. The validated model was then used to derive values for length of queue, waiting time, and proportion of simultaneous retrievals. Finally, the model was run in an extended Monte-Carlo format to establish the relationship of the current system to proposed performance frontiers based on waiting time, simultaneous retrievals, and missed missions. Results: This thesis demonstrated that standard queueing theory could describe the operations of the ScotSTAR systems, describing M/G/1 and M/G/2 queue types for SPRS and EMRS respectively. The DES model based on this was able to accurately replicate the real-world system in retrospective simulation (mean model accuracy: SPRS = 91.0%, EMRS = 91.9%), and partially replicate the contemporaneous state of the system (mean model accuracy: SPRS = 82.2%, EMRS = 89.0%). The model then derived plausible values for length of queue, waiting times, and simultaneous retrieval proportions. Lastly, the model demonstrated a 95th percentile of waiting time (Wq95) of 1 hour for secondary retrievals as being the most significant performance frontier. SPRS was demonstrated to be operating approximately 196 missions per year over this frontier, EMRS had capacity for an extra 26 primary or 23 secondary missions per year before reaching the frontier. Conclusions: Standard queueing theory is able to accurately describe the constituent parameters of the ScotSTAR systems. A discrete event simulation model can, with some limitations, accurately replicate the real-world to allow the derivation of performance descriptors which are unmeasurable in the real-world. Furthermore, such a model can also demonstrate the relationship between the current state of the system and potential performance frontiers

Modelling activities at a neurological rehabilitation unit

A queuing model is developed for the neurological rehabilitation unit at Rookwood Hospital in Cardiff. Arrivals at the queuing system are represented by patient referrals and service is represented by patient length of stay (typically five months). Since there are often delays to discharge, length of stay is partitioned into two parts: admission until date ready for discharge (modelled by Coxian phase-type distribution) and date ready for discharge until ultimate discharge (modelled by exponential distribution). The attributes of patients (such as age, gender, diagnosis etc) are taken into account since they affect these distributions. A computer program has been developed to solve this multi-server (21 bed) queuing system to produce steady-state probabilities and various performance measures. However, early on in the project it became apparent that the intensity of treatment received by patients has an effect on the time, from admission, until they are ready for discharge. That is, the service rates of the Coxian distribution are dependent on the amount of therapy received over time. This directly relates to the amount of treatment allocated in the weekly timetables. For the physiotherapy department, these take about eight hours to produce each week by hand. In order to ask the valuable what-if questions that relate to treatment intensity, it is therefore necessary to produce an automated scheduling program that replicates the manual assignment of therapy. The quality of timetables produced using this program was, in fact, considerably better than its alternative and so replaced the by-hand approach. Other benefits are more clinical time (since less employee input is required)and a convenient output of data and performance measures that are required for audit purposes. Once the model is constructed a number of relevant hypothetical scenarios are considered. Such as, what if delays to discharge are reduced by 50%? Also, through the scheduling program, the effect of changes to the composition of staff or therapy sessions can be evaluated, for example, what if the number of therapists is increased by one third? The effects of such measures are analysed by studying performance measures (such as throughput and occupancy) and the associated costs