In this paper, we investigate a single-server Poisson input Queueing model, wherein arrivals of units are in bulk. The arrival rate of the units is state dependent, and service time is arbitrary distributed. It is also assumed that the system is subject to breakdown, and the failed server immediately joins the repair facility which takes constant duration to repair the server. By using supplementary variable technique, we obtain the probability generating function of the number of units in the system which is further used to establish some performance indices such as the mean number of units in the system, mean waiting time, etc. Special cases are also discussed. In order to obtain approximate values of system state probabilities, the principle of maximum entropy is employed. Numerical results are also presented to validate the analytical formulae.

In this paper, we consider a multi-server queuing system in which additional servers are allowed for a longer Queue to reduce the customer’s balking and reneging behavior. The customers are assumed to arrive in Poisson fashion and their service- times follow exponential distribution. The number of servers in long run depends on the Queue-length. We obtain equilibrium Queue size distribution. The expression for expected number of customers in the system in the long run has been obtained. Some other performance measures have also been provided. In order to match our results with earlier work, special cases are discussed. The numerical results are presented to verify the validity of the proposed analytical method.

Drivers in urban neighborhoods who patrol streets, seeking inexpensive on-street parking create a significant fraction of measured traffic congestion. The pool of drivers patrolling at any time can be modeled as a Queue, where ‘Queue service’ is the act of parking in a recently vacated parking space and Queue discipline is SIRO – Service In Random Order. We develop a Queueing model of such driver behavior, allowing impatient drivers to abandon the Queue and to settle for more expensive off-street parking. We then relate the model to the economic theory of congestion pricing, arguing that price differentials between on-street and off-street parking should be reduced in order to reduce traffic congestion. Reducing the number of "patrolling drivers" often can reduce urban road congestion significantly, in some cases as effectively as technologically expensive road pricing schemes that cordon off the center city.

In this paper we study two echelon multi-server tandom queuing systems where customers arrive according to a poisson process with two different rates. The service rates at both echelons are independent of each other. The service times of customers is assumed to be completed in two stages. The service times at each stage are exponentially distributed. At the first stage, the customers may balk (i.e. reject to join the waiting line) when all servers are busy. The higher echelon has a limited buffer space. The steady state Queue size distribution has been obtained for both stages. We investigate the properties of a Hessenberg matrix which is required for the complete specification of the generating function for ready use.

Waiting in long Queue is undesirable due to which in some practical situations customers become impatient and leave the system. In such cases providing additional servers can reduce long Queue. This investigation deals with multi-server queuing system with additional servers and discouragement. The customers arrive in poisson fashion and are served exponentially by a pool of permanent and additional servers on the basis of FCFS Queue discipline. The explicit formula to determine the number of customers in the Queue has been obtained. Numerical illustration has been provided to validate the analytical results established. Graphs are drawn for expected number of customers in the system and probability of all additional servers being busy in order to visualize the effect of various parameters.

This investigation deals with MX/G/1 Queueing model with setup, bulk-arrival, loss-delay and balking. The provision of second optional service apart from essential service by an unreliable server is taken into consideration. We assume that the delay customers join the Queue when server is busy whereas loss customers depart from the system. After receiving the essential service, the customers may opt for the optional service with some probability or may leave the system. The server is unreliable and hence may breakdown in both essential and optional service cases and requires a setup time before being repaired. The service during essential service, setup times and repair times for both cases are general distributed while the service time during optional service is exponential. Using the supplementary variable technique, the equations governing the model are constructed. The steady state Queue size distribution is then obtained using Laplace transform and probability generating functions. The numerical results for various performance indices have been obtained and illustrated graphically.

The automobile workshop Queue system has been optimized using various approaches, such as queuing theory, simulation, and probability. The utilization of response surface methodology (RSM) for optimizing automobile workshop Queue systems is not yet established. The utilization of RSM with direct observation enables the detection of patterns of correlations between variables and responses, which are then represented through mathematical equations. The optimization process involves numerous factors that impact Queue performance, which can be categorized into two parts. The number of servers, number of phases, number of workers, worker experience, and layout are classified in inner design. This study examines the relationship between two components of the outer design, specifically the arrival rate and the interarrival time. The responses analyzed are Queue cost, service time, average customer waiting time, and number of customers. The findings indicate that Queue costs are not reliable for establishing the optimum value due to the significant impact of the cost structure on the structure of the optimal location. This study discovered that the number of leaving customers is related to Queue costs and is relevant in selecting the optimal point. This study also formulates mathematical equations for predicting the optimal point. This study emphasizes the necessity for further investigation to uncover alternative mathematical equations that can precisely predict the optimal conditions for various types of services.