M/M/1/N Queues with Energy Enabled Service and General Vacation Times

Recently, call centers are important from a customer service point of view because they have a significant impact on customer satisfaction. In general, most of the management cost for call centers is labor cost for Customer Service Representatives (CSRs), and it is important for companies to manage CSRs in a cost-effective manner, keeping a high quality of customer service. Therefore, most of companies install Interactive Voice Response systems (IVRs) in order to not only reduce CSR management cost, but also to provide high-quality customer service. In this paper, focusing on call centers with IVRs, we investigate the impact of the service time at IVRs and number of IVRs on the utilization of a CSR. To this end, we model the call center with IVRs by a queueing system with retrials, analyzing the steady state probability by a continuous-time Markov chain. We derive performance measures such as the mean number of redialing customers, the blocking probability and the mean sojourn time. Numerical results show that under a low arrival rate of new calls, the utilization of a CSR is insensitive to the IVR service time as well as to the number of IVRs, and that the utilization grows proportionally with the increase in the probability that a customer in an IVR leaves for the CSR service.

The subject of discussion in this chapter is the stochastic approach to model investigation of queuing systems (QS) with waiting buffer (queue) and server (service unit). Two types of QS are presented: single-channel and multi-channel QS. Single-channel QS is discussed in the first part of the chapter with presentation of main characteristics such as workload of the resource, queuing length, total number of requests in system, waiting time and time for presence in the system. The second part deals with organization of analytical investigation of QS and in particular presentation of discrete and continuous time Markov chains. The two possibilities for investigation were considered – during the development of a transient regime and the conditions for reaching a steady state regime. Three versions of QS are presented in QS – with an infinite buffer, with a limited buffer, and QS with request rejections in the absence of a buffer. Some examples for stochastic model investigation are presented in the last part.

We study a quantum entanglement switch that serves $k$ users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns.

The purpose of this study is to find out the queuing model using the structure of the single phase multi-channel queue system in customer service at PT Bank BRI Raha Branch and find out how to complete the queue model on customer service at PT Bank BRI Raha Branch. This research was conducted with direct observation on customer service at Bank BRI Raha Branch. From the data obtained, a steady state test and a Chi-Square kindness test were carried out on arrival patterns and service patterns. Then complete the queuing model using the single phase multi channel queue system structure. The results of the analysis showed that the arrival of customers at Bank BRI Raha Branch was 630 customers with an arrival rate of 2 people per hour and a customer service rate of 5 customers per hour. The queuing system in customer service at Bank BRI Raha Branch follows the queue model which means that the Poisson distributed arrival rate and service time are exponential, the number of channels in the dual system (2 customer service with 1 queue line), the queue discipline used by First Come First Serve (those served are customers who arrive first), the number of incoming customers is not limited or infinite in the queuing system and the population size at the input source is infinite. Busy period opportunities as large as , the average number of customers waiting in the queue is person, the average number of customer in the system is person, the average time spent by a customer waiting is hours, the average time spent in the system including services is hours.

A multi-server queueing system with finite and infinite buffers is considered. The input flow is described by the BMAP (Batch Markovian Arrival Process). The service time has the PH (Phase) type distribution. Besides ordinary (positive) customers, the MAP (Markovian Arrival Process) of negative customers arrives to the system. A negative customer can delete an ordinary customer in service if the state (phase) of its PH-service process does not belong to some given set of so-called protected phases. The stationary distribution of the system states and waiting time distribution are derived. The main performance measures of the queueing system considered are computed and numerically illustrated.

This article presents parameter estimation of phase-type (PH) distribution. PH distribution is widely used in model-based performance evaluation such as reliability and queueing system, and is defied by an absorbing continuous-time Markov chain (CTMC). Since non-Markovian model can be approximated by a CTMC by replacing general distributions with PH distributions, the parameter estimation is a challenging issue in stochastic modeling. We focus on the statistical inference algorithms to estimate not only the parameters of PH distribution with grouped, truncated and missing data but also the probability density function, and give two examples of PH fitting in reliability engineering.

This study investigates the queuing system and service efficiency at the Bank BJB sub-branch office in West Bandung Regency, focusing on enhancing customer service operations. With the significant increase in banking sector customers in Indonesia, including a growth in accounts in both conventional and digital financial institutions, the banking industry faces challenges in maintaining speed and efficiency in customer service. Long queues and time-consuming services have led to customer dissatisfaction and reduced operational efficiency​​.The research examines the current performance of the queuing system and teller service time, analyzing the root causes of prolonged waiting and service times, and developing strategies to optimize these times​​. The Bank BJB KCP serves a varied customer base, with complex banking requirements that often lead to longer service times​​. The queuing system employs a multi-channel, single-phase model, and the tellers offer several services, each with its own service level agreement​​.Observational data collected over 12 working days revealed the busiest hours and performance metrics, indicating high congestion during peak hours​​. Root cause analysis, considering both teller and customer perspectives, identified key issues contributing to the queuing system's inefficiency. The study's findings aim to enhance the overall customer service efficiency at Bank BJB KCP.

The M/G/1 classic queueing system was extended by many authors in last two decades. The systems with server’s vacation are important models that extend the M/G/1 queueing system. Also another condition such as admissibility restricted may occur in systems. From this motivation, in this system I consider a single server queue with batch arrival Poisson input. There is a restricted admissibility of arriving batches in which not all batches are allowed to join the sys-tem at all times. At each service completion epoch, the server may apt to take a vacation with probability θ or else with probability 1 ? θ may continue to be available in the system for the next service. The vacation period of the server has two heterogenous phases. Phase one is compulsory, and phase two follows the phase one vacation in such a way that the server may take phase two with probability p or may return back to the system with probability 1 ? p. The vacation times are assumed to be general. All stochastic processes involved in this system (service and vacation times) are inde-pendent of each other. We derive the PGF’s of the system and by using them the informance measures are obtained. Some numerical approaches are examined the validity of results.

In this paper, we analyze an unreliable queueing system consisting of an infinite buffer and two heterogeneous servers. The main server (server 1) is unreliable, while the server 2 is considered as the reserve server and is assumed to be absolutely reliable. The service times have the PH-type (Phase-type) distribution. If both servers are able to provide the service, they serve a customer independently of each other. The service of a customer is completed when his/her service by any of two servers is finished. After the service completion, both servers immediately start the service of the next customer, if he/she presents in the system. If the system is idle, the servers wait for arrival of the new customer. The input flow is described by the BMAP (Batch Markovian Arrival Process). Breakdowns arrive to the server 1 according to a MAP (Markovian Arrival Process). After breakdown occurrence, repair of the server starts. The repair time also has the PH-type (Phase-type) distribution. The customers, which meet the servers busy upon arrival, join a buffer. They will be picked up for the service according to the First-In-First-Out discipline. The customers arrived at the same batch are picked up for the service in random order. If a customer arriving from outside or from a buffer sees only server 2 ready for service while the server 1 is under repair, only server 2 starts the service of this customer. But if server 1 is repaired before service completion of this customer, server 1 immediately begins the service of this customer. For this model, we derive ergodicity condition, calculate the key performance measures of the system and derive an expression for the Laplace-Stieltjes transform of the sojourn time distribution of an arbitrary customer.

Queueing theory or waiting line theory is a theory that deals with the queue process from the customer comes, queue to be served, served and left on service facilities. Queue occurs because of a mismatch between the numbers of customers that will be served with the available number of services, as an example at XYZ insurance company in Tasikmalaya. This research aims to determine the characteristics of the queue system which then to optimize the number of server in term of total cost. The result shows that the queue model can be represented by (M/M/4):(GD/∞/∞), where the arrivals are Poisson distributed while the service time is following exponential distribution. The probability of idle customer service is 2,39% of the working time, the average number of customer in the queue is 3 customers, the average number of customer in a system is 6 customers, the average time of a customer spent in the queue is 15,9979 minutes, the average time a customer spends in the system is 34,4141 minutes, and the average number of busy customer servicer is 3 server. The optimized number of customer service is 5 servers, and the operational cost has minimum cost at Rp 4.323.

In this paper, we consider a tandem dual queuing system consisting of multi-server stages. Stage 1 is characterized by an infinite buffer, one-by-one service of customers, and an exponential distribution of service times. Stage 2 is characterized by a finite buffer and a phase-type distribution of service times. Service at Stage 2 is provided to groups of customers. The service time of a group depends on the size of the group. The size is restricted by two thresholds. The waiting time of a customer at each stage is limited by a random variable with an exponential distribution, with the parameter depending on the stage. After service at Stage 1, a customer can depart from the system or try to enter Stage 2. If the buffer at this stage is full, the customer is either lost or returns for service at Stage 1. Customer arrivals are described by the versatile Markov arrival process. The system is studied via consideration of a multi-dimensional continuous-time Markov chain. Numerical examples, which highlight the influence of the thresholds on the system performance measures, are presented. The possibility of solving optimization problems is illustrated.

We consider a single-server queueing system with server vacations as the important component of the polling queueing model of a real-world system. Period of continuous operation of the server (the maximum server attendance time) is restricted, but the service of a customer cannot be interrupted when this period expires. Such features are inherent for many real-world systems with resource sharing. We assume that the customers arrival is described by the Markovian Arrival Process and service, vacation and maximum server attendance times have a phase-type distribution. We derive the stationary distributions of the system states and waiting time. Taking in mind the necessity of further application of the results to modeling the polling queueing systems, the distribution of the server visiting time is derived. Extensive numerical results are presented. They highlight that an account of the coefficient of variation of vacation and maximum attendance time is very important for exact evaluation of the key performance measures of the system, while only the results for the coefficient of variation equal to zero or one are known in the literature. Impact of the possible customers impatience, which is intuitively important because the time-limited service is considered, is confirmed by the results of the numerical experiments. Optimization problem of matching the durations of vacation and maximum attendance time is considered.

The current paper presented a stochastic integrated queueing-inventory-routing problem into a green dual-channel supply chain considering an online retailer with a vehicle-routing problem (VRP) and a traditional retailing channel with an M/M/C queueing system. A mixed-integer non-linear programming model (MINLP) is presented to address the integrated VRP and M/M/C queueing system. The suggested model makes decisions about optimal routing, delivery time interval to customers, number of servers in traditional retailers, inventory replenishment policies, and retailers’ price. For the first time, this model considers two retailing channels simultaneously under different uncertainty, including demand, delivery lead time, service time, and delivery time interval to customers. The inventory model also follows a continuous-time Markov chain. The small-scale test problems are solved using GAMS software. Since the problem is NP-hard, this study conducts a comprehensive comparative analysis of the performance of 13 different metaheuristics. The ant lion optimiser, dragonfly algorithm, grasshopper optimisation algorithm, Harris-hawks optimisation, moth-flame optimisation algorithm, multi-verse optimizer, sine cosine algorithm, salp-swarm algorithm, the whale optimisation algorithm, grey-wolf optimiser, genetic algorithm, differential evolution, and particle swarm optimization are algorithms that were chosen for this study. Comprehensive statistical tests were conducted to evaluate the performance of these methods. Furthermore, the model is executed for construction material producers as a case study. Finally, sensitivity analyses were conducted on crucial model parameters; and managerial insights were recommended.

We compare the performance of seven methods in computing or approximating service levels for nonstationary M(t)/M/s(t) queueing systems: an exact method (a Runge-Kutta ordinary-differential-equation solver), the randomization method, a closure (or surrogate-distribution) approximation, a direct infinite-server approximation, a modified-offered-load infinite-server approximation, an effective-arrival-rate approximation, and a lagged stationary approximation. We assume an exhaustive service discipline, where service in progress when a server is scheduled to leave is completed before the server leaves. We used all of the methods to solve the same set of 640 test problems. The randomization method was almost as accurate as the exact method and used about half the computational time. The closure approximation was less accurate, and usually slower, than the randomization method. The two infinite-server-based approximations, the effective-arrival-rate approximation, and the lagged stationary approximation were less accurate but had computation times that were far shorter and less problem-dependent than the other three methods.

In the previous chapters we studied queueing systems with different interarrival and service time distributions. Chapter 7 is devoted to the analysis of queueing systems with exponential interarrival and service time distributions. The number of customers in these queueing systems is characterized by CTMCs with a generally nonhomogeneous birth-and-death structure. In contrast, Chap. 8 is devoted to the analysis of queueing systems with nonexponential interarrival and service time distributions. It turns out that far more complex analysis approaches are required for the analysis of queues with nonexponential interarrival and service time distributions. In this chapter we introduce queueing systems whose interarrival and service time distributions are nonexponential, but they can be analyzed with CTMCs. Indeed in this chapter we demonstrate the use of the results of Chap. 5 for the analysis of queueing systems with phase-type (PH) distributed interarrival and service times or with arrival and service processes that are MAPs. The main message of this chapter is that in queueing models the presence of PH or MAP processes instead of exponential distributions results in a generalization of the underlying CTMCs from birth-and-death processes to quasi-birth-and-death (QBDs) processes.