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
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
The U.N.Educational,Scientific and Cultural Organization(UNESCO)works to preserve significant and inspirational places worldwide.Designated World Heritage sites,they are as diverse as Yellowstone National Park,Shark B...The U.N.Educational,Scientific and Cultural Organization(UNESCO)works to preserve significant and inspirational places worldwide.Designated World Heritage sites,they are as diverse as Yellowstone National Park,Shark Bay in Australia and the historic center of Vienna,and they symbolize the world's collective history,culture and landscape.展开更多
Facing various pieces of information disclosed by the system upon arrival,customers often exhibit different strategic responses.In this paper,customers'strategic behavior is studied in a Markovian queue with Berno...Facing various pieces of information disclosed by the system upon arrival,customers often exhibit different strategic responses.In this paper,customers'strategic behavior is studied in a Markovian queue with Bernoulli-type working vacations.Upon completion of a service,the server starts a working vacation if the system is empty.If the system is found to be non-empty,the server takes a working vacation with a certain probability.During a working vacation,the server provides service at a lower service rate.Upon arrival,each customer decides whether to join the system or not based on the information disclosed and a reward-cost structure.The authors study the equilibrium balking strategies of customers at two information levels.For the fully observable case,the authors derive the two-dimensional threshold strategies,under which customers behave accordingly in the regular state and the working vacation state.For the partially observable case,the authors obtain a threshold strategy that completely depends on the queue length of the system.The influence of input parameters on the equilibrium strategies is discussed by numerical examples.Sensitivity analysis shows that reducing the vacation probability or rising the vacation rate will encourage more customers to join the system,thereby improving the system throughput.In addition,the disclosure of real-time server state information will also improve the system throughput.展开更多
This paper studies customer joining behavior and system regulation strategy in nonexhaustive visible M/M/m queues with synchronous vacations of a part of the servers.Once this part of the servers are idle,they take mu...This paper studies customer joining behavior and system regulation strategy in nonexhaustive visible M/M/m queues with synchronous vacations of a part of the servers.Once this part of the servers are idle,they take multiple vacations simultaneously(vacation period).Until there are customers waiting in the queue,they are reactivated and all servers are busy or idle(busy period).The authors call this part of the servers as“partial servers”.In view of the fully visible queue and the almost visible queue,the authors obtain customers’equilibrium joining threshold strategies and their socially optimal joining threshold strategies,respectively,and observe that customer joining behavior in equilibrium generally makes the system overcrowded,which makes the equilibrium social welfare lower than the optimal social welfare.After regulation,interestingly,for optimizing social welfare,the system manager hopes not only customers arriving in vacation period pay attention to the number of partial servers,but also customers arriving in busy period should care about it rather than ignore.Moreover,arranging more servers for vacation does not necessarily lead to the decrease of social welfare on condition that the number of partial servers is close to m.As for the information advantage of the fully visible case,it is not obvious for increasing social welfare and even unfavorable to servers’profit unless the number of partial servers is big enough.Furthermore,given the different composition of social welfare,there exists the optimal number of partial servers and the optimal arrival rate of customers for maximizing social welfare.展开更多
This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. U...This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n^+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0^+, n^+] from any initial state. Meanwhile, the relationship among departure process, server's state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures,including the expected length of server busy period, server's actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N*for minimizing the system cost under a given cost structure.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
A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed （i.i.d.） random variables, that is max{ηi, 1 ≤ i ≤ n}. T...A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed （i.i.d.） random variables, that is max{ηi, 1 ≤ i ≤ n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.展开更多
This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually...This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer＇s ob- servation epochs for partial-batch rejection policy. The blocking probability of the first, an arbitrary- and the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.展开更多
In this paper, we analyze a bulk input M^[X]/M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matr...In this paper, we analyze a bulk input M^[X]/M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function（PGF） of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical M^[X]/M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.展开更多
We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation.Then by using this result we prove that the essential growth bound of the C0...We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation.Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0,the C0-semigroup is not compact,not eventually compact,even not quasi-compact.Moreover,we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution.In addition,we obtain the spectral radius and essential spectral radius of the C0-semigroup.Lastly,we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.展开更多
We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtai...We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.展开更多
This paper studies a cold standby repairable system with working vacations and vacation interruption. The repairman's multiple vacations policy, the working vacations policy and the vacation interruption are consi...This paper studies a cold standby repairable system with working vacations and vacation interruption. The repairman's multiple vacations policy, the working vacations policy and the vacation interruption are considered simultaneously. The lifetime of components follows a phase-type(PH) distribution. The repair time in the regular repair period and the working vacation period follow other two PH distributions at different rates. For this system, the vector-valued Markov process governing the system is constructed. We obtain several important performance measures for the system in transient and stationary regimes applying matrixanalytic methods. Finally, a numerical example is given to illustrate the results obtained.展开更多
This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the mu...This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer＇s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.展开更多
In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process （BMAP） and Markovian arrival proc...In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process （BMAP） and Markovian arrival process （MAP） respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated （multiple） vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.展开更多
In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M（n）/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues wi...In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M（n）/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues with various types of station vacations as special cases. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirement is the Laplace-Stieltjes transform of the interarrival distribution as well as the state-dependent service rate and state-dependent vacation rate. In a subsequent companion paper, we study its dual system M（n）/G/1/K queue with statedependent vacations.展开更多
Using a bounding technique, we prove that the fluid model of generalized Jackson network （GJN） with vacations is the same as a GJN without vacations, which means that vacation mechanism does not affect the dynamic p...Using a bounding technique, we prove that the fluid model of generalized Jackson network （GJN） with vacations is the same as a GJN without vacations, which means that vacation mechanism does not affect the dynamic performance of GJN under fluid approximation. Furthermore, in order to present the impact of vacation on the performance of GJN, we show that exponential rate of convergence for fluid approximation only holds for large N, which is different from a GJN without vacations. The results on fluid approximation and convergence fate are embodied by the queue length, workload, and busy time processes.展开更多
从鲜花供品到静居日,从梯田游览到海浪驰骋,巴厘岛让我们放慢脚步,瞬间发现喜悦。Imagine a vacation that feels like stepping into a reallife adventure game.That's Bali!Its special secret is called"menyama braya,&qu...从鲜花供品到静居日,从梯田游览到海浪驰骋,巴厘岛让我们放慢脚步,瞬间发现喜悦。Imagine a vacation that feels like stepping into a reallife adventure game.That's Bali!Its special secret is called"menyama braya,"which means living peacefully with everything and everyone.展开更多
Hi,I'm Young from Zhangjiagang Yangshe Experimental School,and I'm in Grade Six.I want to share my fantastic summer vacation with you!This summer,I went on a trip to Macao with my family.It was an amazing and ...Hi,I'm Young from Zhangjiagang Yangshe Experimental School,and I'm in Grade Six.I want to share my fantastic summer vacation with you!This summer,I went on a trip to Macao with my family.It was an amazing and unforgettable experience for me!展开更多
Due to its unique geography,Jinan has the largest underground water vein(地下水脉)in China.Thus,with its 72 famous springs,Jinan has become a popular vacation destination.The springs in Jinan,which are the most outsta...Due to its unique geography,Jinan has the largest underground water vein(地下水脉)in China.Thus,with its 72 famous springs,Jinan has become a popular vacation destination.The springs in Jinan,which are the most outstanding features,were first recorded in Li Daoyuan’s Commentary on the Waterways Classic.The text states that there are 72 famous springs in Jinan.However,in reality,the number of springs far exceeds this number,reaching thousands.These springs are not only abundant in quantity,but also diverse in form,showing the wonderful creativity of nature.What’s even more amazing is that springs usually appear in suburban areas.That’s because spring water usually flows from outside the city into the city,while Jinan is completely opposite.Consequently,springs are everywhere in Jinan,including the busiest city center!展开更多
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.
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.
文摘The U.N.Educational,Scientific and Cultural Organization(UNESCO)works to preserve significant and inspirational places worldwide.Designated World Heritage sites,they are as diverse as Yellowstone National Park,Shark Bay in Australia and the historic center of Vienna,and they symbolize the world's collective history,culture and landscape.
基金supported in part by the National Natural Science Foundation of China under Grant No.72371259the Emerging Interdisciplinary Project of Central University of Finance and Economics under Grant No.CUFE-21XXJC010+1 种基金the Project of Establishing the“Double First-Class”Discipline of Surveying and Mapping Science and Technology under Grant No.GCCRC202307the Key Science and Technology Research Project of Henan Province under Grant No.242102210137。
文摘Facing various pieces of information disclosed by the system upon arrival,customers often exhibit different strategic responses.In this paper,customers'strategic behavior is studied in a Markovian queue with Bernoulli-type working vacations.Upon completion of a service,the server starts a working vacation if the system is empty.If the system is found to be non-empty,the server takes a working vacation with a certain probability.During a working vacation,the server provides service at a lower service rate.Upon arrival,each customer decides whether to join the system or not based on the information disclosed and a reward-cost structure.The authors study the equilibrium balking strategies of customers at two information levels.For the fully observable case,the authors derive the two-dimensional threshold strategies,under which customers behave accordingly in the regular state and the working vacation state.For the partially observable case,the authors obtain a threshold strategy that completely depends on the queue length of the system.The influence of input parameters on the equilibrium strategies is discussed by numerical examples.Sensitivity analysis shows that reducing the vacation probability or rising the vacation rate will encourage more customers to join the system,thereby improving the system throughput.In addition,the disclosure of real-time server state information will also improve the system throughput.
基金supported by the National Natural Science Foundation of China under Grant No.71971188the Humanities and Social Science Fund of Ministry of Education of China under Grant No.22YJCZH086the Hebei Natural Science Foundation under Grant Nos.G2023203008 and G2022203003。
文摘This paper studies customer joining behavior and system regulation strategy in nonexhaustive visible M/M/m queues with synchronous vacations of a part of the servers.Once this part of the servers are idle,they take multiple vacations simultaneously(vacation period).Until there are customers waiting in the queue,they are reactivated and all servers are busy or idle(busy period).The authors call this part of the servers as“partial servers”.In view of the fully visible queue and the almost visible queue,the authors obtain customers’equilibrium joining threshold strategies and their socially optimal joining threshold strategies,respectively,and observe that customer joining behavior in equilibrium generally makes the system overcrowded,which makes the equilibrium social welfare lower than the optimal social welfare.After regulation,interestingly,for optimizing social welfare,the system manager hopes not only customers arriving in vacation period pay attention to the number of partial servers,but also customers arriving in busy period should care about it rather than ignore.Moreover,arranging more servers for vacation does not necessarily lead to the decrease of social welfare on condition that the number of partial servers is close to m.As for the information advantage of the fully visible case,it is not obvious for increasing social welfare and even unfavorable to servers’profit unless the number of partial servers is big enough.Furthermore,given the different composition of social welfare,there exists the optimal number of partial servers and the optimal arrival rate of customers for maximizing social welfare.
基金supported by the National Natural Science Foundation of China under Grant Nos.71571127and 71171138
文摘This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n^+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0^+, n^+] from any initial state. Meanwhile, the relationship among departure process, server's state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures,including the expected length of server busy period, server's actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N*for minimizing the system cost under a given cost structure.
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.
基金Supported by the National Natural Science Foundation of China (No. 10826047 and No.10901023)by the Fundamental Research Funds for the Central Universities under Contract BUPT2009RC0707
文摘A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed （i.i.d.） random variables, that is max{ηi, 1 ≤ i ≤ n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.
文摘This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer＇s ob- servation epochs for partial-batch rejection policy. The blocking probability of the first, an arbitrary- and the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.
基金supported by National Natural Science Foundation of China(No. 10671170)Natural Science Foundation of Hebei Province(No. F2008000864)
文摘In this paper, we analyze a bulk input M^[X]/M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function（PGF） of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical M^[X]/M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.
基金the National Natural Science Foundation of China (Grant No. 11961062)。
文摘We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation.Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0,the C0-semigroup is not compact,not eventually compact,even not quasi-compact.Moreover,we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution.In addition,we obtain the spectral radius and essential spectral radius of the C0-semigroup.Lastly,we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.
基金Supported by the National Natural Science Foundation of China(11671204)
文摘We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.
基金supported by the National Natural Science Foundation of China(71371031)
文摘This paper studies a cold standby repairable system with working vacations and vacation interruption. The repairman's multiple vacations policy, the working vacations policy and the vacation interruption are considered simultaneously. The lifetime of components follows a phase-type(PH) distribution. The repair time in the regular repair period and the working vacation period follow other two PH distributions at different rates. For this system, the vector-valued Markov process governing the system is constructed. We obtain several important performance measures for the system in transient and stationary regimes applying matrixanalytic methods. Finally, a numerical example is given to illustrate the results obtained.
文摘This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer＇s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.
基金supported by the National Natural Science Foundation of China (No. 10871064)
文摘In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process （BMAP） and Markovian arrival process （MAP） respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated （multiple） vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.
基金supported by National Science Foundation under DMI-0200306supported in part by a grant from National Natural Science Foundation of China under No.70228001.
文摘In this paper we study a queueing system with state-dependent services and state-dependent vacations, or simply G/M（n）/1/K. Since the service rate is state-dependent, this system includes G/M/c and G/M/c/K queues with various types of station vacations as special cases. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirement is the Laplace-Stieltjes transform of the interarrival distribution as well as the state-dependent service rate and state-dependent vacation rate. In a subsequent companion paper, we study its dual system M（n）/G/1/K queue with statedependent vacations.
文摘Using a bounding technique, we prove that the fluid model of generalized Jackson network （GJN） with vacations is the same as a GJN without vacations, which means that vacation mechanism does not affect the dynamic performance of GJN under fluid approximation. Furthermore, in order to present the impact of vacation on the performance of GJN, we show that exponential rate of convergence for fluid approximation only holds for large N, which is different from a GJN without vacations. The results on fluid approximation and convergence fate are embodied by the queue length, workload, and busy time processes.
文摘从鲜花供品到静居日,从梯田游览到海浪驰骋,巴厘岛让我们放慢脚步,瞬间发现喜悦。Imagine a vacation that feels like stepping into a reallife adventure game.That's Bali!Its special secret is called"menyama braya,"which means living peacefully with everything and everyone.
文摘Hi,I'm Young from Zhangjiagang Yangshe Experimental School,and I'm in Grade Six.I want to share my fantastic summer vacation with you!This summer,I went on a trip to Macao with my family.It was an amazing and unforgettable experience for me!
文摘Due to its unique geography,Jinan has the largest underground water vein(地下水脉)in China.Thus,with its 72 famous springs,Jinan has become a popular vacation destination.The springs in Jinan,which are the most outstanding features,were first recorded in Li Daoyuan’s Commentary on the Waterways Classic.The text states that there are 72 famous springs in Jinan.However,in reality,the number of springs far exceeds this number,reaching thousands.These springs are not only abundant in quantity,but also diverse in form,showing the wonderful creativity of nature.What’s even more amazing is that springs usually appear in suburban areas.That’s because spring water usually flows from outside the city into the city,while Jinan is completely opposite.Consequently,springs are everywhere in Jinan,including the busiest city center!
