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
By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its s...By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its steady-state solution.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 examines an M[x]/G/1 queueing system with an unreliable server and a delayed repair, in which the server operates a randomized vacation policy with multiple vacations. Whenever the system is empty, the serv...This paper examines an M[x]/G/1 queueing system with an unreliable server and a delayed repair, in which the server operates a randomized vacation policy with multiple vacations. Whenever the system is empty, the server immediately takes a,vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 -p. Whenever one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. The server may also meet an unpredictable breakdown and the repair may be delayed. For such a system the authors derive the distributions of some important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle period and the busy period. The authors perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.展开更多
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 National Natural Science Foundation of China (GrantNo. 10861011)
文摘By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its steady-state solution.
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 examines an M[x]/G/1 queueing system with an unreliable server and a delayed repair, in which the server operates a randomized vacation policy with multiple vacations. Whenever the system is empty, the server immediately takes a,vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 -p. Whenever one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. The server may also meet an unpredictable breakdown and the repair may be delayed. For such a system the authors derive the distributions of some important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle period and the busy period. The authors perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.
