Queue Length Simulations in a Finite Single-line Queueing System with Repeated Calls

2000 Mathematics Subject Classification: 60K25.

On the Busy Period in One Finite Queue of M/G/1 Type with Inactive Orbit

The paper deals with a single server finite queuing system where the customers, who failed to get service, are temporarily blocked in the orbit of inactive customers. This model and its variants have many applications, especially for optimization of the corresponding models with retrials. We analyze the system in non-stationary regime and, using the discrete transformations method study, the busy period length and the number of successful calls made during it. ACM Computing Classification System (1998): G.3, J.7.

Better queue management in a busy public hospital of a developing country without appointment system::an application using data envelopment analysis

Queuing is a key efficiency criterion in any service industry, including Healthcare. Almost all queue management studies are dedicated to improving an existing Appointment System. In developing countries such as Pakistan, there are no Appointment Systems for outpatients, resulting in excessive wait times. Additionally, excessive overloading, limited resources and cumbersome procedures lead to over-whelming queues. Despite numerous Healthcare applications, Data Envelopment Analysis (DEA) has not been applied for queue assessment. The current study aims to extend DEA modelling and demonstrate its usefulness by evaluating the queue system of a busy public hospital in a developing country, Pakistan, where all outpatients are walk-in; along with construction of a dynamic framework dedicated towards the implementation of the model. The inadequate allocation of doctors/personnel was observed as the most critical issue for long queues. Hence, the Queuing-DEA model has been developed such that it determines the ‘required’ number of doctors/personnel. The results indicated that given extensive wait times or length of queue, or both, led to high target values for doctors/personnel. Hence, this crucial information allows the administrators to ensure optimal staff utilization and controlling the queue pre-emptively, minimizing wait times. The dynamic framework constructed, specifically targets practical implementation of the Queuing-DEA model in resource-poor public hospitals of developing countries such as Pakistan; to continuously monitor rapidly changing queue situation and display latest required personnel. Consequently, the wait times of subsequent patients can be minimized, along with dynamic staff scheduling in the absence of appointments. This dynamic framework has been designed in Excel, requiring minimal training and work for users and automatic update features, with complex technical aspects running in the background. The proposed model and the dynamic framework has the potential to be applied in similar public hospitals, even in other developing countries, where appointment systems for outpatients are non-existent.

Estimation of the Traffic Intensity in a Piecewise-stationary Mt/Gt/1 Queue with Probing

We use a probing strategy to estimate the time dependent traffic intensity in an Mt/Gt/1 queue, where the arrival rate and the general service-time distribution change from one time interval to another, and derive statistical properties of the proposed estimator. We present a method to detect a switch from a stationary interval to another using a sequence of probes to improve the estimation. At the end, we compare our results with two estimators proposed in the literature for the M/G/1 queue.

Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

In the Hammersley-Aldous-Diaconis process, infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x, whose nearest neighbor to the right is at y, jumps at rate y - x to a position uniformly distributed in the interval (x, y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate, and the (attempted) services with rate rho > lambda Then put first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n - 1 queues in tandem with n - 1 priority types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the reversed process. The coupled process is a transformation of the multi-line process and its invariant measure is the transformation described above of the product measure.

POLLING SYSTEMS WITH PARAMETER REGENERATION, THE GENERAL CASE

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the sth moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447-1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space.

Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process

Quasi-birth-and-death (QBD) processes with infinite “phase spaces” can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the “phase” giving the state of the first queue and the “level” giving the state of the second queue. In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of Neuts’s R-matrix and show that the decay rate of the stationary distribution of the “level” process is not always equal to the convergence norm of R. In fact, we show that we can obtain any decay rate from a certain range by controlling only the transition structure at level zero, which is independent of R. We also consider the sequence of tandem queues that is constructed by restricting the waiting room of the first queue to some finite capacity, and then allowing this capacity to increase to infinity. We show that the decay rates for the finite truncations converge to a value, which is not necessarily the decay rate in the infinite waiting room case. Finally, we show that the probability that the process hits level n before level 0 given that it starts in level 1 decays at a rate which is not necessarily the same as the decay rate for the stationary distribution.

A fast cross-entropy method for estimating buffer overflows in queuing networks

In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for the M/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.

An improved seeded region growing algorithm

Recently Adams and Bischof (1994) proposed a novel region growing algorithm for segmenting intensity images. The inputs to the algorithm are the intensity image and a set of seeds - individual points or connected components - that identify the individual regions to be segmented. The algorithm grows these seed regions until all of the image pixels have been assimilated. Unfortunately the algorithm is inherently dependent on the order of pixel processing. This means, for example, that raster order processing and anti-raster order processing do not, in general, lead to the same tessellation. In this paper we propose an improved seeded region growing algorithm that retains the advantages of the Adams and Bischof algorithm fast execution, robust segmentation, and no tuning parameters - but is pixel order independent. (C) 1997 Elsevier Science B.V.

Formal derivation of finite state machines for class testing

Previous work on generating state machines for the purpose of class testing has not been formally based. There has also been work on deriving state machines from formal specifications for testing non-object-oriented software. We build on this work by presenting a method for deriving a state machine for testing purposes from a formal specification of the class under test. We also show how the resulting state machine can be used as the basis for a test suite developed and executed using an existing framework for class testing. To derive the state machine, we identify the states and possible interactions of the operations of the class under test. The Test Template Framework is used to formally derive the states from the Object-Z specification of the class under test. The transitions of the finite state machine are calculated from the derived states and the class's operations. The formally derived finite state machine is transformed to a ClassBench testgraph, which is used as input to the ClassBench framework to test a C++ implementation of the class. The method is illustrated using a simple bounded queue example.

Experimental evidence that partner choice is a driving force in the payoff distribution among cooperators or mutualists: the cleaner fish case

Supply and demand largely determine the price of goods on human markets. It has been proposed that in animals, similar forces influence the payoff distribution between trading partners in Sexual selection, intraspecific cooperation and interspecific mutualism. Here we present the first experimental evidence supporting biological market theory in it study on cleaner fish, Labroides dimidiatus. Cleaners interact with two classes of clients: choosy client species with access to several cleaners usually do not queue for service and do not return if ignored, while resident client species with access to only one cleaning station do queue or return. We used plexiglas plates with equal amounts of food to stimulate these behaviours of the two client classes. Cleaners soon inspected 'choosy' plates before 'resident' plates. This supports previous field observations that suggest that client species with access to several cleaners exert choice to receive better(immediate) service.

Divorce, dispersal and incest avoidance in the cooperatively breeding superb fairy-wren Malurus cyaneus

1. Between 1988 and 2001, we studied social relationships in the superb fairy-wren Malurus cyaneus (Latham), a cooperative breeder with male helpers in which extra-group fertilizations are more common than within-pair fertilizations. 2. Unlike other fairy-wren species, females never bred on their natal territory. First-year females dispersed either directly from their natal territory to a breeding vacancy or to a foreign 'staging-post' territory where they spent their first winter as a subordinate. Females dispersing to a foreign territory settled in larger groups. Females on foreign territories inherited the territory if the dominant female died, and were sometimes able to split the territory into two by pairing with a helper male. However, most dispersed again to obtain a vacancy. 3. Females dispersing from a staging post usually gained a neighbouring vacancy, but females gaining a vacancy directly from their natal territory travelled further, perhaps to avoid pairing or mating with related males. 4. Females frequently divorced their partner, although the majority of relationships were terminated by the death of one of the pair. If death did not intervene, one-third of pairings were terminated by female-initiated divorce within 1000 days. 5. Three divorce syndromes were recognized. First, females that failed to obtain a preferred territory moved to territories with more helpers. Secondly, females that became paired to their sons when their partner died usually divorced away from them. Thirdly, females that have been in a long relationship divorce once a son has gained the senior helper position. 6. Dispersal to avoid pairing with sons is consistent with incest avoidance. However, there may be two additional benefits. Mothers do not mate with their sons, so dispersal by the mother liberates her sons to compete for within-group matings. Further, divorcing once their son has become a breeder or a senior helper allows the female to start sons in a queue for dominance on another territory. Females that do not take this option face constraints on their ability to recruit more sons into the local neighbourhood.

Analysing TDMA with slot skipping

We propose a schedulability analysis for a particular class of time division multiple access (TDMA) networks, which we label as TDMA/SS. SS stands for slot skipping, reflecting the fact that a slot is skipped whenever it is not used. Hence, the next slot can start earlier in benefit of hard real-time traffic. In the proposed schedulability analysis, we assume knowledge of all message streams in the system, and that each node schedules messages in its output queue according to a rate monotonic policy (as an example). We present the analysis in two steps. Firstly, we address the case where a node is only permitted to transmit a maximum of one message per TDMA cycle. Secondly, we generalise the analysis to the case where a node is assigned a budget of messages per TDMA cycle it may transmit. A simple algorithm to assign budgets to nodes is also presented.

From task scheduling in single processor environments to message scheduling in a PROFIBUS fieldbus network

In this paper we survey the most relevant results for the prioritybased schedulability analysis of real-time tasks, both for the fixed and dynamic priority assignment schemes. We give emphasis to the worst-case response time analysis in non-preemptive contexts, which is fundamental for the communication schedulability analysis. We define an architecture to support priority-based scheduling of messages at the application process level of a specific fieldbus communication network, the PROFIBUS. The proposed architecture improves the worst-case messages’ response time, overcoming the limitation of the first-come-first-served (FCFS) PROFIBUS queue implementations.

Supporting real-time parallel task models with work-stealing

Dynamic parallel scheduling using work-stealing has gained popularity in academia and industry for its good performance, ease of implementation and theoretical bounds on space and time. Cores treat their own double-ended queues (deques) as a stack, pushing and popping threads from the bottom, but treat the deque of another randomly selected busy core as a queue, stealing threads only from the top, whenever they are idle. However, this standard approach cannot be directly applied to real-time systems, where the importance of parallelising tasks is increasing due to the limitations of multiprocessor scheduling theory regarding parallelism. Using one deque per core is obviously a source of priority inversion since high priority tasks may eventually be enqueued after lower priority tasks, possibly leading to deadline misses as in this case the lower priority tasks are the candidates when a stealing operation occurs. Our proposal is to replace the single non-priority deque of work-stealing with ordered per-processor priority deques of ready threads. The scheduling algorithm starts with a single deque per-core, but unlike traditional work-stealing, the total number of deques in the system may now exceed the number of processors. Instead of stealing randomly, cores steal from the highest priority deque.