A transient solution to the M/M/c queuing model equation with balking and catastrophes

In this paper, we consider a Markovian multi-server queuing system with balking and catastrophes. The probability generating function technique along with the Bessel function properites is used to obtain a transient solution to the queuing model. The transient probabilities for the number of customers in the system are obtained explicitly. The expressions for the time-dependent expected number of customers in the system are also obtained. Finally, applications of the model are also discussed

Transient behavior of / /1 [ ] M G X Queueing Model With State Dependent Arrival, Balking and Bernoulli Vacation

This paper analyzes single server queueing system wherein the arrival units are in batches following compound Poisson process with varying arrival rates for different states. The arrival rates differ for different states like idle state, busy state and vacation period. The server provides service one by one and after completion of a service, the server either takes a vacation of random length with probability p (0 ï‚£ p ï‚£1) or continue to serve the next customer with probability 1-p, if any. Both service time and vacation time follow general distribution. In addition to this one of the customer impatience, balking is also discussed. We have obtained time dependent solution followed by the steady state analysis with some performance measures of our model

On impatience in M/M/1/N/DWV queue with vacation interruption

In this paper, we establish a cost optimization analysis for an M/M/1/N queuing system with differentiated working vacations, Bernoulli schedule vacation interruption, balking and reneging. Once the system is empty, the server waits a random amount of time before he goes on working vacation during which service is provided with a lower rate. At the instant of the service achievement in the vacation period, if there are customers present in the system, the vacation is interrupted and the server returns to the regular busy period with probability β\u27 or continues the working vacation with probability 1 - β\u27. Whenever the working vacation is ended, the server comes back to the normal busy period. If the system is empty, the server can take another working vacation of shorter duration. In addition, it is supposed that during both busy and working vacation periods, arriving customers may become impatient with individual timers exponentially distributed. Explicit expressions for the steady-state system size probabilities are derived using recursive technique. Further, interesting performance measures are explicitly obtained. Then, we construct a cost model in order to determine the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Finally, numerical illustrations are added to validate the theoretical results

Stability Condition of a Retrial Queueing System with Abandoned and Feedback Customers

This paper deals with the stability of a retrial queueing system with two orbits, abandoned and feedback customers. Two independent Poisson streams of customers arrive to the system, and flow into a single-server service system. An arriving one of type i; i = 1; 2, is handled by the server if it is free; otherwise, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. The customer in the orbit either attempts service again after a random time or gives up receiving service and leaves the system after a random time. After the customer is served completely, the customer will decide either to join the retrial group again for another service or leave the system forever with some probability

Capacity flexibility of a maintenance service provider in specialized and commoditized system environments

In the last decades, after-sales services have become increasingly important since service is a source of differentiation as well as a lucrative business opportunity due to the substantial amount of revenue that can be generated from the products in use throughout their life cycle. Following this trend, many after-sales service providers have emerged in the market or evolved as semi-autonomous units within the OEM (Original Equipment Manufacturer) companies. In this thesis, we focus on the maintenance aspect of after-sales services. We assume that a maintenance service provider (MSP) is running a repair shop in an environment with numerous operating systems that are prone to failure. The MSP is responsible for keeping all systems in an environment up and working. We mainly focus on two types of environments: 1) Specialized System Environment 2) Commoditized System Environment. The systems in the first environment are highly customized. They are designed and built specifically following the owners’ precise requirements. Defense systems, specific lithography systems, mission aircrafts or other advanced/complex, engineer-to-order capital goods are examples of such specialized systems. Due to the diversity of owners’ requirements, each system develops many unique characteristics, which make it hard, if not impossible, to find a substitute for the system, in the market as a whole. In the second environment, the systems are more generic in terms of their functionality. Trucks, cranes, printers, copy machines, forklifts, computer systems, cooling towers, some common medical devices (i.e. anesthesia, x-ray and ultrasound machines, etc…), power systems are examples of such more commoditized systems. Due to the more generic features of the owners’ requirements, it is easier to find a substitute for a system in the market, with more or less the same functionality, for short-term hiring purposes. Upon a system breakdown, the defective unit (system/subsystem) is sent to the repair shop. MSP is responsible for the repair and also liable for the costs related to the down time. In order to alleviate the down-time costs, there are chiefly two different downtime service strategies that the MSP can follow, depending on the environment the repair shop is operating in. In the specialized system environment, the MSP holds a spare unit inventory for the critical subsystem that causes most of the failures. The downtime service related decision in such a case would be the inventory level of the critical spare subsystems. On the other hand, in the commoditized system environment, rather than keeping a spare unit inventory, the MSP hires a substitute system from an agreed rental store/3rd party supplier. The downtime service related decision in this case is the hiring duration. Next to the above downtime service related decisions, repair shop’s capacity level is the other primary determinant of the systems’ uptime/availability. Since maintenance is a labor-intensive industry, the capacity costs constitute a large portion of the total costs. Increasing pressure on profitability and the growing role of External Labor Supplier Agencies motivate service provider firms to scrutinize the prospects and possibilities of capacity flexibility by using contingent workforce. For various reasons, flexible capacity practices in real life are often periodic, and the period length is both a decision parameter and a metric for flexibility. A shorter period length implies more frequent adapting possibilities and a better tailoring of the capacity. On the other hand, the flexible capacity cost per unit time is higher for shorter period lengths due to the compensating wage differentials, which models the relation between the wage rate and the unpleasantness, risk or other undesirable attributes of the job. Certainly, short period length in this context is an undesirable attribute for the flexible capacity resource, as it mandates the resource to switch tasks and to be ready/available more frequently, without the guarantee that s/he will be actually employed. Therefore, we propose several empirically testable functional forms for the cost rate of a flexible capacity unit, which are decreasing with the period length and, in the limit, approaches to the cost rate of a permanent capacity unit from above. In the light of discussions above, we investigate three different capacity modes in this dissertation: ¿ Fixed Capacity Mode: In this mode, all of the capacity is permanent and ready for use in the repair shop. This mode serves as a reference point in order to assess the benefits of other flexible capacity modes. The relevant capacity decision in this mode is the single capacity level of the repair shop. ¿ Periodic Two-Level Capacity Mode: In this mode, we assume two levels of repair shop capacity: permanent and permanent plus contingent capacity levels. The permanent capacity is always available in the system, whereas the deployment of the contingent capacity is decided at the start of each period based on the number of units waiting to be repaired in the shop. The relevant capacity decisions in this mode are the permanent and contingent capacity levels, the period length and the states (in terms of number of defective units waiting) where the contingent capacity is deployed. ¿ Periodic Capacity Sell-Back Mode: In this mode, the failed units are sent to the repair shop at regular intervals in time. Due to this admission structure, when the repair of all the defective units in the repair shop are completed in a period, it is known that no new defective parts will arrive to the shop at least until the start of the next period. This certainty in idle times allows for a contract, where the repair shop capacity is sold at a reduced price to the capacity agency where it is assigned to other tasks until the start of the next period. The original cost of the multi-skilled repair shop capacity per time unit is higher than the permanent capacity cost that is mentioned in previous modes due to the compensation factors such as additional skills, frequent task switching and transportation/transaction costs. Similar to the previous capacity mode, the compensation decreases with the length of the period length. The relevant capacity decisions in this mode are the capacity level and the period length. The primary goal of this thesis is to develop quantitative models and methods for taking optimal capacity decisions for the repair shop in the presence of the capacity modes described above and to integrate these decisions with the other downtime service decisions of the MSP for two different types of system environments (specialized vs. commoditized). After the introduction of the problem, concepts and literature review are given in Chapters 1. In Chapter 2, we focus on the use of capacity flexibility in the repair operations of the MSP in specialized system environment. The capacity related decisions are integrated with the decision on the stock level of the spare unit inventory for all three capacity modes. In Chapter 3 we investigate the same three capacity modes in a (partially) commoditized system environment, where hiring a substitute system for a pre-determined, uniform duration becomes the conventional method upon a failure. In this chapter the decision on the hiring duration is integrated with the other capacity related decisions. Then we provide some preliminary analysis and give the early results on the hybrid strategy where both "keeping stock" and "hire substitute" strategies are followed. Finally in Chapter 4, we summarize our results, give the conclusion and discuss the topics covered in this thesis with a brief exploration on the future research. The numerical results reveal that, in both specialized and commoditized system environments, substantial cost savings (up to 70%) can be achieved under periodic two-level capacity and periodic capacity sell-back modes compared to the fixed capacity mode. However, both period length and the compensation scheme of the capacity resources greatly influence the savings, even in some cost instances, flexible modes (periodic two-level and capacity sell-back) become less economical compared to the fixed capacity mode. Cost parameter instances in which each of the 3 capacity modes becomes cost-optimal, the characteristics of the cost savings and the sensitivity analysis of cost/policy parameters are investigated in both of the system environments in Chapter 2 and Chapter 3, respectively. In the commoditized system environment, under the same cost parameter settings, the hiring substitute from an external supplier for a fixed duration causes a better, more refined and certain control compared to keeping an inventory. Hybrid strategy, in which a substitute is hired after a stock-out instance, is applicable in commoditized as well as commoditizing (previously specialized systems that are in the ongoing commoditization process) system environments. Hybrid strategy outperforms both "only keeping stock" and "only hiring substitute" alternatives; however, in the commoditized system environment, a MSP may still have a proclivity to employ the "hiring substitute" strategy only, because it does not require any initial investment, which is convenient for SMEs. These issues will be explicated further in Chapter 5. We believe that the framework, the design and analysis of the problems addressed as well as the results and the insights obtained in this dissertation can help and motivate other researchers/practitioners to further investigate the cost saving prospects from capacity flexibility in maintenance service operations. We also anticipate that the commoditization framework described in this thesis will be increasingly useful in the future, since the commoditization of the parts/machines will be much more widespread, pushing all the after-sales service providers to compete on the efficiency of their operatio