In this paper, we study an M/M/1 queue with multiple working vacations under following Bernoulli control policy: at the instants of the completion of a service in vacation, the server will interrupt the vacation and e...In this paper, we study an M/M/1 queue with multiple working vacations under following Bernoulli control policy: at the instants of the completion of a service in vacation, the server will interrupt the vacation and enter regular busy period with probability 1 p (if there are customers in the queue) or continue the vacation with probability p. For this model, we drive the analytic expression of the stationary queue length and demonstrate stochastic decomposition structures of the stationary queue length and waiting time, also we obtain the additional queue length and the additional delay of this model. The results we got agree with the corresponding results for working vacation model with or without vacation interruption if we set p = 0 or p = 1, respectively.展开更多
This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient q...This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival Mx/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.展开更多
We study a batch arrival MX/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular trans...We study a batch arrival MX/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular transition probability matrix of two-dimensional Markov chain and matrix analytic method, the probability generating function (PGF) of the stationary system length distribution is obtained, from which we obtain the stochastic decomposition structure of system length which indicates the relationship with that of the MX/M/1 queue without vacation. Some performance indices are derived by using the PGF of the stationary system length distribution. It is important that we obtain the Laplace Stieltjes transform (LST) of the stationary waiting time distribution. Further, we obtain the mean system length and the mean waiting time. Finally, numerical results for some special cases are presented to show the effects of system parameters.展开更多
This article examines the effects of reneging, server breakdown and server vacation on the various states of the batch arrivals queueing system with single server providing service to customers in three fluctuating mo...This article examines the effects of reneging, server breakdown and server vacation on the various states of the batch arrivals queueing system with single server providing service to customers in three fluctuating modes. In this queueing system, any batch arrival joins the queue if the server is busy or on vacation or under repair. However, if the server is free, one customer from the arriving batch joins the service immediately while others join the queue. In case of server breakdown, the customer whose service is interrupted returns back to the head of the queue. As soon as the server has is repaired, the server attends to the customer in mode 1. For this queueing system, customers that are impatient due to breakdown and server vacation may renege (leave the queue without getting service). Due to fluctuating modes of service delivery, the system may provide service with complete or reduced efficiency. Consequently, we construct the mathematical model and derive the probability generating functions of the steady state probabilities of several states of the system including the steady state queue size distribution. Further, we discuss some particular cases of the proposed queueing model. We present numerical examples in order to demonstrate the effects of server vacation and reneging on the various states of the system. The study revealed that an increase in reneging and a decrease in server vacation results in a decrease in server utilization and an increase in server’s idle time provided rates of server breakdown and repair completion are constant. In addition, the probability of server vacation, the probability of system is under repair and the probabilities that the server provides service in three fluctuating modes decreases due to an increase in reneging and a decrease in vacation completion rates.展开更多
We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the...We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the server commences service and serves exhaustively before taking another vacation. If the server finds the system empty, it waits a fixed time c. At the expiration of this time, the server commences another vacation if no customer has arrived;otherwise, it serves exhaustively before commencing another vacation. Analytical results are derived for the mean waiting time in the system. The timeout scheme is shown to be a generalized scheme of which both the single vacation and multiple vacations schemes are special cases, with c=∞and c=0, respectively. The model is extended to the N-policy vacation queueing system.展开更多
In China,mountain destinations offer more opportunities for vacation than beaches and springs,with a larger national coverage,richer landscapes and various diverse folk cultures.Hunan Province,for example,serves as a ...In China,mountain destinations offer more opportunities for vacation than beaches and springs,with a larger national coverage,richer landscapes and various diverse folk cultures.Hunan Province,for example,serves as a typical mountain tourism destination.Our research aims to assess the potential suitability and component pattern of holidaymaking in Hunan Province to support for scientific endeavors and sustainable use of mountain resources in Hunan.An approach for mapping potential suitability was constructed based on Weighted Linear Combination(WLC)of six primary factors including terrain,climate suitability period,biology and environment,human activity culture,hydrological landscape,and road infrastructure,which is calculated on the base of eleven secondary factors used as suitability criteria.Analytic hierarchy process(AHP)and geographic information system(GIS)were also integrated into the evaluation models to prepare six single and complex factor suitability maps for mountain vacation.Multivariate clustering was used to clarify the component pattern of single factor combinations in each mountain vacation suitability index(MVSI)class.The results have shown that:1)Regions with high suitability are distributed with characteristics of significant dispersion and limited aggregation;2)15.89%of the sample mountains are highly suitable,30.78%are moderately suitable,42.61%are marginally suitable and 10.72%are poorly suitable;3)The complexity of index combination corresponding to each level is different;4)Different combinations emphasize different destination needs.The study reveals the suitability class,characteristics,and limiting factors of mountain vacation in different geographical locations on the provincial scale.These elements play an instructive role in scientific decision-making and orderly exploitation of resources among Hunan’s Mountain destinations.Furthermore,the study can also be used as a basis for the suitability evaluation of mountain vacation in other areas.展开更多
We study a vacation queueing system with a single server simultaneously dealing with an M/G/1 and an M/D/1 queue. Two classes of units, priority and non-priority, arrive at the system in two independent Poisson stream...We study a vacation queueing system with a single server simultaneously dealing with an M/G/1 and an M/D/1 queue. Two classes of units, priority and non-priority, arrive at the system in two independent Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority units and a deterministic service to the non-priority units. We further assume that the server may take a vacation of random length just after serving the last priority unit present in the system. We obtain steady state queue size distribution at a random epoch. Corresponding results for some special cases, including the known results of the M/G/1 and the M/D/1 queues, have been derived.展开更多
Puqian Town, Heyuan City has beautiful natural environment and agricultural landscape and unique hot spring resources, so it is very suitable for the development of rural vacation industry. At present, there were only...Puqian Town, Heyuan City has beautiful natural environment and agricultural landscape and unique hot spring resources, so it is very suitable for the development of rural vacation industry. At present, there were only two vacation projects of Bavaria Manor and Ketianxia Hot Spring Resort in the town, and the radiation driving effect is not obvious. In this paper, literature research and interview methods were used to collect data. After 15 interviewees were effectively interviewed and hundreds of data were collected, the PEST theory model and balanced points card theory were used to measure the development factors of rural vacation in Puqian Town, Heyuan City. SWOT matrix matching analysis method was adopted to determine the development strategies of local rural vacation industry.展开更多
I have a summer vacation plan.I’m going to the USA with my family.Because it’s very far from Zhongshan,I’m going to America by plane.First, I’d like to go to Washington D.C.There, I will be able to see the White H...I have a summer vacation plan.I’m going to the USA with my family.Because it’s very far from Zhongshan,I’m going to America by plane.First, I’d like to go to Washington D.C.There, I will be able to see the White House.It’s said that it is very beautiful.But I only saw it on TV or in the newspapers or on the magazines before.When I get there, I’ll take some nice photos.When I come back to Zhongshan, I’ll share them with my classmates.展开更多
Meaningful Summer Vacation I'll be a senior high school student after this summer vacation, so my parents and I decided making①this summer vacation a special and memorable one for me.
In this study,the authors consider an M/M/1 queuing system with attached inventory under an(s,S)control policy.The server takes multiple vacations whenever the inventory is depleted.It is assumed that the lead time an...In this study,the authors consider an M/M/1 queuing system with attached inventory under an(s,S)control policy.The server takes multiple vacations whenever the inventory is depleted.It is assumed that the lead time and the vacation time follow exponential distributions.The authors formulate the model as a quasi-birth-and-dearth(QBD)process and derive the stability condition of the system.Then,the stationary distribution in product form for the joint process of the queue length,the inventory level,and the server’s status is obtained.Furthermore,the conditional distributions of the inventory level when the server is on and operational,and when it is off due to a vacation,are derived.Using the stationary distribution,the authors obtain some performance measures of the system.The authors investigate analytically the effect of the server’s vacation on the performance measures.Finally,several numerical examples are presented to investigate the effects of some parameters on the performance measures,the optimal policy,and the optimal cost.展开更多
In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operati...In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operative phase j,j=1,K¯,customers are served one by one.Once the system is empty,the servers have to wait a random period of time before leaving,causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate.Using the method of the probability generating functions,we establish the steady-state analysis of the system.Special cases of the queueing model are presented.Then,explicit expressions of the useful system characteristics are derived.In addition,a cost model is constructed to define the optimal values of service rates,simultaneously,to minimize the total expected cost per unit time via a quadratic fit search method.Numerical examples are provided to display the impact of different system characteristics.展开更多
In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation ...In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.展开更多
In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate ...In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate of the customer is lower than a normal busy period.If the system is non-empty when the server returns from the working vacation,the server resumes the normal service period.Otherwise,the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next arrival.Closed-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions,and the performance measures of the system are derived using generating functions.The stochastic decomposition structures of the mean queue length and mean waiting time are verified.The effects of the system parameters on some performance measures had been numerically illustrated.展开更多
Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying ...Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying the well-known stochastic decomposition property of the steady-state queue size,the probability generating function of the steady-state queue length distribution is obtained.Moreover,the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations.Finally,employing the renewal reward theorem,the explicit expression of the long-run expected cost per unit time is given.Furthermore,we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N,D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.展开更多
基金Foundation item: Supported by the National Science Foundation of China(60874083) Supported by the 2011 National Statistical Science Development Funds(2011LY014) Supported by the 2012 Soft Science Devel- opment Funds of Science and Technology Committee of Henan Province(122400450090)
文摘In this paper, we study an M/M/1 queue with multiple working vacations under following Bernoulli control policy: at the instants of the completion of a service in vacation, the server will interrupt the vacation and enter regular busy period with probability 1 p (if there are customers in the queue) or continue the vacation with probability p. For this model, we drive the analytic expression of the stationary queue length and demonstrate stochastic decomposition structures of the stationary queue length and waiting time, also we obtain the additional queue length and the additional delay of this model. The results we got agree with the corresponding results for working vacation model with or without vacation interruption if we set p = 0 or p = 1, respectively.
基金the National Outstanding Youth Science Foundation !(79725002) the Youth Science Foundation of UEST.
文摘This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival Mx/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.
文摘We study a batch arrival MX/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular transition probability matrix of two-dimensional Markov chain and matrix analytic method, the probability generating function (PGF) of the stationary system length distribution is obtained, from which we obtain the stochastic decomposition structure of system length which indicates the relationship with that of the MX/M/1 queue without vacation. Some performance indices are derived by using the PGF of the stationary system length distribution. It is important that we obtain the Laplace Stieltjes transform (LST) of the stationary waiting time distribution. Further, we obtain the mean system length and the mean waiting time. Finally, numerical results for some special cases are presented to show the effects of system parameters.
文摘This article examines the effects of reneging, server breakdown and server vacation on the various states of the batch arrivals queueing system with single server providing service to customers in three fluctuating modes. In this queueing system, any batch arrival joins the queue if the server is busy or on vacation or under repair. However, if the server is free, one customer from the arriving batch joins the service immediately while others join the queue. In case of server breakdown, the customer whose service is interrupted returns back to the head of the queue. As soon as the server has is repaired, the server attends to the customer in mode 1. For this queueing system, customers that are impatient due to breakdown and server vacation may renege (leave the queue without getting service). Due to fluctuating modes of service delivery, the system may provide service with complete or reduced efficiency. Consequently, we construct the mathematical model and derive the probability generating functions of the steady state probabilities of several states of the system including the steady state queue size distribution. Further, we discuss some particular cases of the proposed queueing model. We present numerical examples in order to demonstrate the effects of server vacation and reneging on the various states of the system. The study revealed that an increase in reneging and a decrease in server vacation results in a decrease in server utilization and an increase in server’s idle time provided rates of server breakdown and repair completion are constant. In addition, the probability of server vacation, the probability of system is under repair and the probabilities that the server provides service in three fluctuating modes decreases due to an increase in reneging and a decrease in vacation completion rates.
文摘We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the server commences service and serves exhaustively before taking another vacation. If the server finds the system empty, it waits a fixed time c. At the expiration of this time, the server commences another vacation if no customer has arrived;otherwise, it serves exhaustively before commencing another vacation. Analytical results are derived for the mean waiting time in the system. The timeout scheme is shown to be a generalized scheme of which both the single vacation and multiple vacations schemes are special cases, with c=∞and c=0, respectively. The model is extended to the N-policy vacation queueing system.
基金supported by the National Key Research and Development Program (No.2019YFD1100404)A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (No.18C1559)+1 种基金Scientific Research Project of Hunan Radio and TV University (No.XDK2019-C-2)Scientific Innovation Fund for Post-graduates of Central South University of Forestry and Technology (CX202102001).
文摘In China,mountain destinations offer more opportunities for vacation than beaches and springs,with a larger national coverage,richer landscapes and various diverse folk cultures.Hunan Province,for example,serves as a typical mountain tourism destination.Our research aims to assess the potential suitability and component pattern of holidaymaking in Hunan Province to support for scientific endeavors and sustainable use of mountain resources in Hunan.An approach for mapping potential suitability was constructed based on Weighted Linear Combination(WLC)of six primary factors including terrain,climate suitability period,biology and environment,human activity culture,hydrological landscape,and road infrastructure,which is calculated on the base of eleven secondary factors used as suitability criteria.Analytic hierarchy process(AHP)and geographic information system(GIS)were also integrated into the evaluation models to prepare six single and complex factor suitability maps for mountain vacation.Multivariate clustering was used to clarify the component pattern of single factor combinations in each mountain vacation suitability index(MVSI)class.The results have shown that:1)Regions with high suitability are distributed with characteristics of significant dispersion and limited aggregation;2)15.89%of the sample mountains are highly suitable,30.78%are moderately suitable,42.61%are marginally suitable and 10.72%are poorly suitable;3)The complexity of index combination corresponding to each level is different;4)Different combinations emphasize different destination needs.The study reveals the suitability class,characteristics,and limiting factors of mountain vacation in different geographical locations on the provincial scale.These elements play an instructive role in scientific decision-making and orderly exploitation of resources among Hunan’s Mountain destinations.Furthermore,the study can also be used as a basis for the suitability evaluation of mountain vacation in other areas.
文摘We study a vacation queueing system with a single server simultaneously dealing with an M/G/1 and an M/D/1 queue. Two classes of units, priority and non-priority, arrive at the system in two independent Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority units and a deterministic service to the non-priority units. We further assume that the server may take a vacation of random length just after serving the last priority unit present in the system. We obtain steady state queue size distribution at a random epoch. Corresponding results for some special cases, including the known results of the M/G/1 and the M/D/1 queues, have been derived.
文摘Puqian Town, Heyuan City has beautiful natural environment and agricultural landscape and unique hot spring resources, so it is very suitable for the development of rural vacation industry. At present, there were only two vacation projects of Bavaria Manor and Ketianxia Hot Spring Resort in the town, and the radiation driving effect is not obvious. In this paper, literature research and interview methods were used to collect data. After 15 interviewees were effectively interviewed and hundreds of data were collected, the PEST theory model and balanced points card theory were used to measure the development factors of rural vacation in Puqian Town, Heyuan City. SWOT matrix matching analysis method was adopted to determine the development strategies of local rural vacation industry.
文摘I have a summer vacation plan.I’m going to the USA with my family.Because it’s very far from Zhongshan,I’m going to America by plane.First, I’d like to go to Washington D.C.There, I will be able to see the White House.It’s said that it is very beautiful.But I only saw it on TV or in the newspapers or on the magazines before.When I get there, I’ll take some nice photos.When I come back to Zhongshan, I’ll share them with my classmates.
文摘Meaningful Summer Vacation I'll be a senior high school student after this summer vacation, so my parents and I decided making①this summer vacation a special and memorable one for me.
基金supported in part by the Natural Science Foundation of China under Grant No.71971189the Natural Science Foundation of Hebei Province under Grant No.A2019203313+1 种基金the Key Project of Scientific Research in Higher Education of Hebei Province of China under Grant No.ZD2018042in part by MEXT,Japan。
文摘In this study,the authors consider an M/M/1 queuing system with attached inventory under an(s,S)control policy.The server takes multiple vacations whenever the inventory is depleted.It is assumed that the lead time and the vacation time follow exponential distributions.The authors formulate the model as a quasi-birth-and-dearth(QBD)process and derive the stability condition of the system.Then,the stationary distribution in product form for the joint process of the queue length,the inventory level,and the server’s status is obtained.Furthermore,the conditional distributions of the inventory level when the server is on and operational,and when it is off due to a vacation,are derived.Using the stationary distribution,the authors obtain some performance measures of the system.The authors investigate analytically the effect of the server’s vacation on the performance measures.Finally,several numerical examples are presented to investigate the effects of some parameters on the performance measures,the optimal policy,and the optimal cost.
文摘In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operative phase j,j=1,K¯,customers are served one by one.Once the system is empty,the servers have to wait a random period of time before leaving,causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate.Using the method of the probability generating functions,we establish the steady-state analysis of the system.Special cases of the queueing model are presented.Then,explicit expressions of the useful system characteristics are derived.In addition,a cost model is constructed to define the optimal values of service rates,simultaneously,to minimize the total expected cost per unit time via a quadratic fit search method.Numerical examples are provided to display the impact of different system characteristics.
基金the National Natural Science Foundation of China(No.61773014)。
文摘In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.
文摘In this paper,we consider the impatient customers in M/M/1 queueing model under variant working vacation policy.The customer’s impatience is due to its arrival during a working vacation period,where the service rate of the customer is lower than a normal busy period.If the system is non-empty when the server returns from the working vacation,the server resumes the normal service period.Otherwise,the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next arrival.Closed-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions,and the performance measures of the system are derived using generating functions.The stochastic decomposition structures of the mean queue length and mean waiting time are verified.The effects of the system parameters on some performance measures had been numerically illustrated.
基金supported by the National Natural Science Foundation of China(No.71571127)the National Natural Science Youth Foundation of China(No.72001181).
文摘Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying the well-known stochastic decomposition property of the steady-state queue size,the probability generating function of the steady-state queue length distribution is obtained.Moreover,the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations.Finally,employing the renewal reward theorem,the explicit expression of the long-run expected cost per unit time is given.Furthermore,we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N,D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.