Fluid Limits for Multiclass Many-Server Queues with General Reneging Distributions and Head-of-the-Line Scheduling

We describe a fluid model with time-varying input that approximates a multiclass many-server queue with general reneging distribution and multiple customer classes (specifically, the multiclass G/GI/N+GI queue). The system dynamics depend on the policy, which is a rule for determining when to serve a given customer class. The class of admissible control policies are those that are head-of-the-line (HL) and nonanticipating. For a sequence of many-server queues operating under admissible HL control policies and satisfying some mild asymptotic conditions, we establish a tightness result for the sequence of fluid scaled queue state descriptors and associated processes and show that limit points of such sequences are fluid model solutions almost surely. The tightness result together with the characterization of distributional limit points as fluid model solutions almost surely provides a foundation for the analysis of particular HL control policies of interest. We leverage these results to analyze a set of admissible HL control policies that we introduce, called weighted random buffer selection (WRBS), and an associated WRBS fluid model that allows multiple classes to be partially served in the fluid limit (which is in contrast to previously analyzed static priority policies).

On A Single Server Queue with Two-Stage First Essential Service Followed by One of the Two Types of Additional Optional Service and Optional Deterministic Server Vacations

We study the steady state behavior of a batch arrival single server queue in which the first service consisting of two stages with general service times G1 and G2 is compulsory. After completion of the two stages of the first essential service, a customer has the option of choosing one of the two types of additional service with respective general service times G1 and G2 . Just after completing both stages of first essential service with or without one of the two types of additional optional service, the server has the choice of taking an optional deterministic vacation of fixed (constant) length of time. We obtain steady state probability generating functions for the queue size for various states of the system at a random epoch of time in explicit and closed forms. The steady state results of some interesting special cases have been derived from the main results.

Usage of pedestrian overpass in Akure, Nigeria

Background This study addressed the safety challenge, motivation, and utilization factors regarding the usage of the pedestrian overpass by different people of dissimilar ages alongside Oba Adeshida road at Oja-Oba (Oba market) in Akure, Ondo State, Nigeria. It addressed the following objectives: determined the value of the pedestrian overpass as used by people of different ages; examined the reasons for using the pedestrian overpass in Akure; and examined the performance level of a pedestrian overpass in Akure. Methods In total, 384 users (who used the bridge) were purposively sampled for questionnaire administration, but 324 users participated in the questionnaire, which included multiple-choice and open-ended questions. Also, personal observation was employed for queue analysis. The “Semantic Differential Scale” and single-server queue system for the single queue were employed for data analysis. Conclusions In Akure, there was low or no usage of the pedestrian overpass when the road median was made use of metal and low concrete; but when the road median was made use of high concrete barrier, there was high usage of the pedestrian overpass. It also revealed that the engineering design of the pedestrian overpass is low compared to the contemporary design solutions in addressing the urban societal needs. The pedestrian overpass was used frequently and every day because it provides access to the car park, market, and central mosque. The performance of the pedestrian overpass in Akure is 105% which indicates that the pedestrian overpass is over-utilized by a pedestrian. It is recommended that there is a need to construct another pedestrian overpass to enhance the efficiency of the structure. The existing and future pedestrian overpass should be redesigned to constitute interesting aesthetics and architectural solutions that could harmonize with the environment and enhance flow of the disabled persons.

Analysis of a bulk arrival N-policy queue with two-service genre, breakdown, delayed repair under bernoulli vacation and repeated service policy.

This article deals with an unreliable bulk arrival single server queue rendering two-heterogeneous optional repeated service (THORS) with delayed repair, under Bernoulli Vacation Schedule (BVS) and N-policy. For this model, the joint distribution of the server's state and queue length are derived under both elapsed and remaining times. Further, probability generating function (PGF) of the queue size distribution along with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, numerical experiments have been carried out in support of the theory.

Analysis of Multi-Server Queue with Self-Sustained Servers

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

On optimal stochastic jumps in multi server queue with impatient customers via stochastic control

<p style='text-indent:20px;'>In this paper, a queuing system as multi server queue, in which customers have a deadline and they request service from a random number of identical severs, is considered. Indeed there are stochastic jumps, in which the time intervals between successive jumps are independent and exponentially distributed. These jumps will be occurred due to a new arrival or situation change of servers. Therefore the queuing system can be controlled by restricting arrivals as well as rate of service for obtaining optimal stochastic jumps. Our model consists of a single queue with infinity capacity and multi server for a Poisson arrival process. This processes contains deterministic rate <inline-formula><tex-math id="M1">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and exponential service processes with <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> rate. In this case relevant customers have exponential deadlines until beginning of their service. Our contribution is to extend the Ittimakin and Kao's results to queueing system with impatient customers. We also formulate the aforementioned problem with complete information as a stochastic optimal control. This optimal control law is found through dynamic programming.</p>