On transient queue-size distribution in the batch arrival system with the N-policy and setup times

In the paper the $M^{X}/G/1$ queueing system with the $N$-policy and setup times is considered. An explicit formula for the Laplace transform of the transient queue-size distribution is derived using the approach consisting of few steps. Firstly, a "special\u27\u27 modification of the original system is investigated and, using the formula of total probability, the analysis is reduced to the case of the corresponding system without limitation in the service. Next, a renewal process generated by successive busy cycles is used to obtain the general result. Sample numerical computations illustrating theoretical results are attached as well

Analysis of batch arrival bulk service queue with multiple vacation closedown essential and optional repair

The objective of this paper is to analyze an queueing model with multiple vacation, closedown, essential and optional repair. Whenever the queue size is less than , the server starts closedown and then goes to multiple vacation. This process continues until at least customer is waiting in the queue. Breakdown may occur with probability when the server is busy. After finishing a batch of service, if the server gets breakdown with a probability , the server will be sent for repair. After the completion of the first essential repair, the server is sent to the second optional repair with probability . After repair (first or second) or if there is no breakdown with probability , the server resumes closedown if less than ` \u27 customers are waiting. Otherwise, the server starts the service under the general bulk service rule. Using supplementary variable technique, the probability generating function of the queue size at an arbitrary time is obtained for the steady-state case. Also some performance measures and cost model are derived. Numerical illustrations are presented to visualize the effect of various system parameters

Analysis of Batch Arrival Single and Bulk Service Queue with Multiple Vacation Closedown and Repair

In this paper, we analyze batch arrival single and bulk service queueing model with multiple vacation, closedown and repair. The single server provides single service if the queue size is ‘\u3c a’ and bulk service if the queue size is ‘ ≥ a’. After completing the service (single or bulk), the server may breakdown with probability ξ and then it will be sent for repair. When the system becomes empty or the server is ready to serve after the repair but no one is waiting, the server resumes closedown and then goes for a multiple vacation of random length. Using supplementary variable technique, the steady-state probability generating function (PGF) of the queue size at an arbitrary time is obtained. The performance measures and cost model are also derived. Numerical illustrations are presented to visualize the effect of system parameters

Energy-saving policies for temperature-controlled production systems with state-dependent setup times and costs

There are numerous practical examples of production systems with servers that require heating in order to process jobs. Such production systems may realize considerable energy savings by temporarily switching off the heater and building up a queue of jobs to be processed later, at the expense of extra queueing costs. In this paper, we optimize this trade-off between energy and queueing costs. We model the production system as an M/G/1 queue with a temperature-controlled server that can only process jobs if a minimum production temperature is satisfied. The time and energy required to heat a server depend on its current temperature, hence the setup times and setup costs for starting production are state dependent. We derive the optimal policy structure for a fluid queue approximation, called a wait-heat-clear policy. Building upon these insights, for the M/G/1 queue we derive exact and approximate costs for various intuitive types of wait-heat-clear policies. Numerical results indicate that the optimal wait-heat-clear policy yields average cost savings of over 40% compared to always keeping the server at the minimum production temperature. Furthermore, an encouraging result for practice is that simple heuristics, depending on the queue length only, have near-optimal performance

Discrete Time Analysis of Consolidated Transport Processes

Diese Arbeit beschäftigt sich mit der Entwicklung zeitdiskreter Modelle zur Analyse von Transportbündelungen. Mit den entwickelten Modellen für Bestands- und Fahrzeugbündelungen, insbesondere Milkrun-Systeme, kann eine detaillierte Leistungsbewertung in kurzer Zeit durchgeführt werden. Darüber hinaus erlauben die Modelle die Analyse der Umschlagslagerbündelungen, beispielweise Hub-und-Spoke-Netzwerke, indem sie im Rahmen einer Netzwerkanalyse mit einander verknüpft werden