Olga S. Dudina

Retrial queuing system with Markovian arrival flow and phase-type service time distribution

We consider a multi-server queuing system with retrial customers to model a call center. The flow of customers is described by a Markovian arrival process (MAP). The servers are identical and independent of each other. A customers service time has a phase-type distribution (PH). If all servers are busy during the customer arrival epoch, the customer moves to the buffer with a probability that depends on the number of customers in the system, leaves the system forever, or goes into an orbit of infinite size. A customer in the orbit tries his (her) luck in an exponentially distributed arbitrary time. During a waiting period in the buffer, customers can be impatient and may leave the system forever or go into orbit. A special method for reducing the dimension of the system state space is used. The ergodicity condition is derived in an analytically tractable form. The stationary distribution of the system states and the main performance measures are calculated. The problem of optimal design is solved numerically. The numerical results show the importance of considering the MAP arrival process and PH service process in the performance evaluation and capacity planning of call centers.

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.

Call center operation model as a MAP/PH/N/R-N system with impatient customers

We analyze a multiserver queueing system with a finite buffer and impatient customers. The arrival customer flow is assumed to be Markovian. Service times of each server are phase-type distributed. If all servers are busy and a new arrival occurs, it enters the buffer with a probability depending on the total number of customers in the system and waits for service, or leaves the system with the complementary probability. A waiting customer may become impatient and abandon the system. We give an algorithm for finding the stationary distribution of system states and derive formulas for basic performance characteristics. We find Laplace-Stieltjes transforms for sojourn and waiting times. Numeric examples are given.

Tandem queueing system with impatient customers as a model of call center with Interactive Voice Response

A tandem queueing system with a Markovian Arrival Process (MAP) useful in modeling a call center with Interactive Voice Response (IVR) is investigated. The first stage has a finite number of servers without buffer while the second stage of the tandem has a finite buffer and a finite number of servers. The service time at the first (second) stage has an exponential (phase type) distribution. A special approach for reducing the number of states of the stochastic process that describes the behavior of the system is used. The main performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution is derived. The numerical results are presented.

Analysis of an mmap/ph1, ph2/n/∞ queueing system operating in a random environment

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

MMAP|M|N queueing system with impatient heterogeneous customers as a model of a contact center

Abstract A multi-server queueing system with infinite buffer and impatient heterogeneous customers as a model of a contact center that processes incoming calls (priority customers) and e-mail requests (non-priority customers) is investigated. The arrival flow is described by a Marked Markovian Arrival Process (MMAP). The service time of priority and non-priority customers by a server has an exponential distribution with different parameters. The steady state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transforms of the sojourn and waiting time distribution are derived. The problem of optimal choice of the number of contact center agents under the constraint that the average waiting time of e-mail requests does not exceed a predefined value is numerically solved.

Hysteresis control by the number of active servers in queueing system MMAP/PH/N with priority service

Abstract The problem of choosing the optimal hysteresis strategy of control by the number of active servers in the multi-server queue is considered. Customers of two types arrive to the system according to the marked Markovian arrival process ( M M A P ). Type 1 customers have a non-preemptive priority, but the buffer for these customers is finite. The buffer for type 2 customers is infinite. The service time distribution is of phase-type ( P H ) depending on the type of customers. Some servers are always active. The rest of servers can be switched on or off depending on the number of customers in the system. The strategy of control by the number of active servers is of hysteresis type. Such a strategy is defined by two sets of thresholds. The servers are activated or switched off depending on the relation of the number of customers and the thresholds. The main contribution of the paper is development of a procedure for computation of the stationary distribution of the system states and the value of economical cost criterion under any fixed thresholds. Numerical results show effectiveness of the hysteresis control and importance of account of correlation in the arrival process and variance of service times.

Priority retrial queueing model operating in random environment with varying number and reservation of servers

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 customers have preemptive priority over type 2 customers. Type 1 customer is lost only if, at its arrival moment, all servers are busy by type 1 customers. The service times of both types of customers have an exponential distribution with the rate depending on the type of a customer. Type 2 customer is accepted for service if the number of busy servers at its arrival epoch does not exceed a fixed threshold. Otherwise, the arriving customer makes a randomized choice to leave the system permanently (to balk) or to join so called orbit and try to obtain service later. The inter-retrial times have an exponential distribution. Customers in orbit can be impatient and may leave the system after an exponentially distributed amount of time. Due to preemptive priority of type 1 customers, service of type 2 customer can be interrupted. In this case, the interrupted customer makes a randomized choice to leave the system permanently or go into orbit. When the random environment changes its state, immediately the following parameters of the system change their value: the total number of servers, the number of servers available for type 2 customers, the matrices defining the arrival process, the rates of service of customers, the rate of retrials, the impatience intensity, the probability of balking the system at the arrival moment or the moment of termination of service of type 2 customer. Behavior of the system is described by the level dependent multi-dimensional Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed via the stationary state probabilities. Numerical illustrations are presented.

Queueing system MAP / M / N as a model of call center with call-back option

A multi-server queueing system with a Markovian Arrival Process (MAP), an infinite buffer and impatient customers useful in modeling a call center with a call-back option is investigated. The service time of a customer by a server has an exponential distribution. If all servers are busy at a customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, the real customer can be impatient and can leave the system without the service or request for call-back (becomes a virtual customer). An efficient algorithm for calculating the stationary probabilities of system states is proposed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution for virtual customers is derived. Some numerical results are presented.

Analysis of Priority Retrial Queue With Many Types of Customers and Servers Reservation as a Model of Cognitive Radio System

Cognitive radio is emerging as one of the key information transmission technologies to enhance spectrum efficiency for dramatically increased wireless network capacity requested by end users. Dynamic spectrum access allows effective use of radio frequency and prevents its underutilization in many real-world networks. It enables unlicensed users to temporarily “borrow” unused spectrum while ensuring that the rights of the incumbent license holders are respected. Problems of optimization of joint access of primary and secondary users can be effectively solved by means of queueing theory. In this paper, the analysis of a novel queueing model suitable for the optimization of access is implemented under quite general assumptions about the system parameters. There are several types of primary customers having different requirements for the service time and preemptive priority over secondary customers. Secondary customers can share a server, while primary customers occupy the whole server. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of phase-type. Effect of retrials of secondary customers is taken into account. An effective way for the analysis of multi-server queues with many types of customers and heterogeneous requirements to the service process is provided and applied.