Arrival Epochs Research Articles

Retrial queueing system with balking, optional service and vacation

In this study, we propose a single server retrial queueing system with balking, second optional service and single vacation. At the arrival epoch, if the server is busy, the arriving job join the orbit or balks the system whereas if the server is free, then the arriving job starts its service immediately. For each job, the server provides two phases of service. All the jobs demand the first essential service, whereas only some of the jobs demand for the second optional service. If the system is empty, then the server becomes inactive and begins a single vacation. If server comes back from the vacation, it does not go for another vacation even if the system is still empty at that time. The steady state distributions of the server state and the number of jobs in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analyzed numerically. A general decomposition law for this retrial queueing system is established.

Two-node fluid network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Two-node fluid network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case ofstrong stabilitywhere both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Optimal Deployment of Emergency Supply Inventory for Recurring Disasters with a Humanitarian Relief Objective

We consider the pre-positioning of emergency supply inventories (ESIs) for recurring disasters such as floods and hurricanes, and investigate the optimal allocation policy with the objective of minimizing the expected total unmet demand. Different from the previous literature that has considered only the static processes of ESI requirements and deployment, in this study, we model them as dynamic processes: the arrival epochs of the ESI at the disaster-affected location(s) from the supply network are part of a dynamic process and the corresponding demand for the ESI is a time-dependent stochastic process. We characterize the matching process between the delivery of the ESI after the landfall of the disaster and the corresponding demand arrival process to derive the total unmet demand.Based on a stylized two-location example, we show that it can be optimal to pre-position more ESI at locations with a smaller disaster-affected population size and/or with a smaller inventory survival rate. The optimal allocation depends on the demand arrival process, lead time and pre-positioned inventory survival rate. We then apply our model to a large-scale humanitarian relief operation case, as in Rawls and Turnquist (2010). Based on the data provided, we compute the optimal pre-positioning decisions for three types of ESIs: water, food, and medical supplies. We find that both the optimal ESI pre-positioned at a particular location and the number of locations that pre-position a positive amount of ESI may decrease with the level of total available ESI. We also find that the optimal ESI pre-positioned at a location can decrease in its own inventory survival rate. Last, we compare our case results with those computed via the static model. We find that our model and algorithms can substantially improve the effectiveness of the ESI pre-positioning.

Fluid Limits for Bandwidth-Sharing Networks with Rate Constraints

Bandwidth-sharing networks as introduced by Roberts and Massoulié [Roberts JW, Massoulié L (1998) Bandwidth sharing and admission control for elastic traffic. Proc. ITC Specialist Seminar, Yokohama, Japan], Massoulié and Roberts [Massoulié L, Roberts JW (1999) Bandwidth sharing: Objectives and algorithms. Proc. IEEE Infocom. (Books in Statistics, New York), 1395–1403] model the dynamic interaction among an evolving population of elastic flows competing for several links. With policies based on optimization procedures, such models are of interest both from a queueing theory and operations research perspective. In the present paper, we focus on bandwidth-sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows. This regime is standard in practice. In particular, we extend previous work by Reed and Zwart [Reed J, Zwart B (2010) Limit theorems for bandwidth-sharing networks with rate constraints. Revised, preprint http://people.stern.nyu.edu/jreed/Papers/BARevised.pdf ] on fluid approximations for such networks: we allow interarrival times, flow sizes, and patient times (i.e., abandonment times measured from the arrival epochs) to be generally distributed, rather than exponentially distributed. We also develop polynomial-time computable fixed-point approximations for stationary distributions of bandwidth-sharing networks, and suggest new techniques for deriving these types of results.

Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option

A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the 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, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented.

A Computational approach to optimized appointment scheduling

Appointment scheduling is prevalent in various healthcare settings. Generally, the objective is to determine a schedule (i.e., the sequence of epochs at which the individual patients are asked to appear) that appropriately balances the interests of the patients (low waiting times) and the medical staff (low idle times). In queueing language, the planner is given the distributions of the service times of the individual clients, and then it is his task to determine the arrival epochs of the clients. In this paper, we demonstrate how to generate schedules that have certain optimality properties. As a general principle, we express the performance of a schedule in terms of its associated utility, which incorporates both waiting times and idle times. In a first class of schedules (referred to as the simultaneous approach), the arrival epochs are chosen such that the sum of the utilities of all clients as well as the service provider is minimized. In a second class (sequential approach), the arrival epoch of the next client is scheduled, given the scheduled arrival epochs of all previous clients. For general service times the numerical evaluation of the optimal schedules is often prohibitive; it essentially requires knowledge of the waiting-time distribution in an appropriately chosen D/G/1 queue. In this paper, we demonstrate that by using the phase-type counterparts of the service-time distributions, it is feasible to efficiently determine an optimized schedule, that is, we obtain accurate results with low computational effort. We do so both for transient scenarios (in which the number of clients is relatively low, so that the interarrival time is not uniform) and stationary scenarios (with many clients, and essentially constant interarrival times). Our approach is backed by several examples, that give insight in the impact of the variability of the service times on the schedule; it also shows the impact of the utility function selected.

Optimizing cloud utilization via switching decisions

This paper studies a control problem for optimal switching on and off a cloud computing services modeled by an M=M=1 queue with holding, running and switching costs. The main result is that an average-optimal policy either always runs the system or is an (M; N)- policy defined by two thresholds M and N, such that the system is switched on upon an arrival epoch when the system size accumulates to N and it is switched off upon a departure epoch when the system size decreases to M. We compare the optimal (M; N)-policy with the classical (0; N)-policy and show the non-optimality of it.

Analysis of Multiserver Queueing System with Opportunistic Occupation and Reservation of Servers

We consider a multiserver queueing system with two input flows. Type-1 customers have preemptive priority and are lost during arrival only if all servers are occupied by type-1 customers. If all servers are occupied, but some provide service to type-2 customers, service of type-2 customer is terminated and type-1 customer occupies the server. If the number of busy servers is less than the thresholdMduring type-2 customer arrival epoch, this customer is accepted. Otherwise, it is lost or becomes a retrial customer. It will retry to obtain service. Type-2 customer whose service is terminated is lost or moves to the pool of retrial customers. The service time is exponentially distributed with the rate dependent on the customer’s type. Such queueing system is suitable for modeling cognitive radio. Type-1 customers are interpreted as requests generated by primary users. Type-2 customers are generated by secondary or cognitive users. The problem of optimal choice of the thresholdMis the subject of this paper. Behavior of the system is described by the multidimensional Markov chain. Its generator, ergodicity condition, and stationary distribution are given. The system performance measures are obtained. The numerical results show the effectiveness of considered admission control.

Analysis and optimization of Guard Channel Policy in cellular mobile networks with account of retrials

A multi-server retrial queue with two types of calls (handover and new calls) is analyzed. This queue models the operation of a cell of a mobile communication network. Calls of two types arrive at the system according to the Marked Markovian Arrival Process. Service times of both types of the calls are exponentially distributed with different service rates. Handover calls have priority over new calls. Priority is provided by means of reservation of several servers of the system exclusively for service of handover calls. A handover call is dropped and leaves the system if all servers are busy at the arrival epoch. A new call is blocked if all servers available to new calls are busy. Such a call has options to balk (to leave the system without getting the service) or to retry later on. The behavior of the system is described by the four-dimensional Markov chain belonging to the class of the asymptotically quasi-Toeplitz Markov chains (AQTMC). In the paper, a constructive ergodicity condition for this chain is derived and the effective algorithm for computing the stationary distribution is presented. Based on this distribution, formulas for various performance measures of the system are obtained. Results of numerical experiments illustrating the behavior of key performance measures of the system depending on the number of the reserved servers under the different shares of the handover and the new calls are presented. An optimization problem is considered and high positive effect of server's reservation is demonstrated.

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 customer’s 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.

Idle time utilization through service to customers in a retrial queue maintaining high system reliability*

We consider a k-out-of-n system in which life times of components are exponentially distributed with parameter λ/i, when there are i operational components. There is a single server who repairs the failed components. In addition, service is rendered to external customers also when there are no failed components (main customers). The external customers arrive according to a BM AP. If the arriving batch of external customers finds a free server, one among them gets into service and others, if any, move to an orbit of infinite capacity. If an arriving batch sees a busy server, the whole batch moves to the orbit. The inter-retrial times are exponentially distributed with parameter αi when there are i customers in the orbit. The external customer gets service only when the server is idle and its service is assumed to be nonpreemptive. The service times of main and external customers follow arbitrary distributions B1(· )a ndB2(·), respectively. The stability condition and steady-state distribution are obtained. Some performance measures are computed and numerical illustrations provided. In this paper we discuss the reliability of a k-out-of-n system subject to repair of failed components by a server in a retrial queue. A k-out-of-n system is characterized by the fact that the system operates as long as there are at least k operational components. We assume that the k-out-of-n system is cold. The system is cold in the sense that operational components do not fail while the system is in a down state (number of failed components at that instant is n − k + 1). Using the same analysis as employed in this paper, one can study the warm and hot systems also (a k-out-of- n system is called a hot system if operational components continue to deteriorate at the same rate while the system is down as when it is up. The system is warm if the deterioration rate while the system is up differs from that when it is down). A repair facility, consisting of a single server, repairs the failed components one at a time. The life-times of components are independent and exponentially distributed random variables with parameter λ/i when i components are operational. Thus on average, λ failures take place in unit time when the system operates with i components. The failed components are sent to the repair facility and are repaired one at a time. The waiting space has capacity to accommodate a maximum of n − k + 1 units in addition to the unit undergoing service. Service times of main customers (components of the k-out-of-n system) are independent identically distributed random variables with distribution function B1(t). In addition to repairing failed components of the system, the repair facility provides service to external customers. However these customers are entertained only when the server is idle (no component of the main system is in repair nor even waiting). These customers are not allowed to use the waiting space at the repair facility. So when external customers arrive for service (arrival process is BM AP) while the server is busy serving a component of the system or an external customer, they are directed to an orbit and try their luck after a random length of time, exponentially distributed with parameter αi when there are i customers in orbit. For a brief description of BM AP we refer to [12] and for an extensive literature survey to [3]. We stress the fact that at the instant when an external customer undergoes service if a component of the system fails, the latter’s repair starts only on completion of service of the external customer. That is, external customers are provided nonpreemptive service. The service times of external customers are independent identically distributed random variables with distribution function B2(t). Since external arrivals form a BM AP, either all in an arriving batch will proceed to an orbit on encountering a busy server or else one among the customers in the batch proceeds for service and the rest are directed to the orbit if the server is idle at that arrival epoch.

Single server unreliable queueing model with removable service station

This investigation deals with a single server queueing model with a removable service station under steady-state conditions. The server is in a controllable position, i.e., the organiser can turn the single server on at any arrival epoch or turn off at any service completion. The number of arrivals follows a Poisson process and the service time follows an exponential distribution. The arrival rates of the customers depend on the various states of the system at any time t. The transient analysis for the queue size distribution has been provided by solving a set of governing equations characterising the model in terms of Laplace transform of state probabilities. Several system characteristics are evaluated by using queue size distribution.

BLOCK-LEVEL SUCCESS AND LOSS RUNS FOR STREAMING SERVICES OVER WIRED-WIRELESS NETWORKS

This paper considers the characteristics of block-level success and loss runs for streaming services over wired-wireless networks in which forward error correction (FEC) is employed for packet-loss resilience. We develop a single-server queueing model where the packet-transmission process over a wireless link is modeled by a Markovian service process (MSP) with two states Good and Bad. The queueing system has two input processes: a general renewal input process and a Poisson one. The former represents a streaming-packet flow, while the latter is for background traffic multiplexed at the queue. We investigate the dynamics of the system state during the interval from the arrival epoch of the first packet of the block to that of the last one, deriving the block-loss (resp. block-success) run distribution, i.e., the probability distribution of the number of consecutive block loss (resp. success). The analytical model is validated by trace-driven simulation experiments. Numerical examples show that when the traffic intensity is high, the moments of block-level loss and success runs are significantly affected by the volume of background traffic and the FEC redundancy, rather than the system capacity.

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N-policy GI/Geo/1 queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.

Performance Analysis of a Single Server Retrial Queueing System with Bernoulli and Phase Type Vacations

This paper analyses the steady state behavior of an M/G/1 retrial queueing system with Bernoulli and phase type vacations. Customers arrive one by one at the system in a Poisson stream. At the arrival epoch, if the server is busy then the arriving customer joins the orbit. If the server is free, then the arriving customer starts its service immediately. The service time of a customer is assumed to be general. At each service completion epoch, the server may opt to take a phase 1 vacation with probability p or else with probability 1- p stay in the system for the next service. After the completion of phase 1 vacation the server may take phase 2 vacation with probability q or return back to the system with probability 1-q. The vacation times are assumed to be general. The service times and vacation times are independent of each other. Generating function technique is applied to obtain the system size and orbit size. Numerical examples are provided to illustrate the sensitivity of the performance measures for changes in the parametric of the system.

Unfinished work for the queue under [formula omitted]-policy with incomplete information on service times

We will consider the queueing model under D-policy with incomplete information on service times. The operator does not know the actual service time of an arriving customer at each arrival epoch, but the correlated amount of the service time is known. We derive the Laplace–Stieltjes transform of the unfinished work and then obtain the mean and variance of the unfinished work. In numerical examples, the distribution function of the unfinished work is calculated by numerically inverting the Laplace–Stieltjes transform. The mean and variance of the unfinished work are depicted.

Queueing system [formula omitted] with impatient heterogeneous customers as a model of call center

A multi-server queueing system with a Markovian arrival process (MAP), a finite buffer and impatient heterogeneous customers useful in modeling a call center is investigated. The servers are identical and independent of each other. The service time of a customer has a Phase type distribution (PH). If all servers are busy at a customer arrival epoch, the customer moves to the buffer with probability, which depends on the number of customers in the system, or, alternatively, leaves the system forever. During a waiting period the customers can be impatient and leave the system. An efficient algorithm for calculating the stationary probabilities of system states is proposed. A special approach for reducing the system dimension is used. Some key performance measures are calculated. The Laplace–Stieltjes transforms of the sojourn and waiting time distribution are derived. Some illustrative numerical results are presented.

A multi-class discrete-time queueing system under the FCFS service discipline

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn’t actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into ‘during the same time-slot’. In other words, it doesn’t specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multi-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ⩽ p ⩽ 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.