A Markovian Working Vacation Queue with Server State Dependent Arrival Rate and with Unreliable Server

We consider single-server scheduling to minimize holding costs where the capacity, or rate of service, depends on the number of jobs in the system, and job sizes become known upon arrival. In general, this is a hard problem, and counter-intuitive behavior can occur. For example, even with linear holding costs the optimal policy may be something other than SRPT or LRPT, it may idle, and it may depend on the arrival rate. We first establish an equivalence between our problem of deciding which jobs to serve when completed jobs immediately leave, and a problem in which we have the option to hold on to completed jobs and can choose when to release them, and in which we always serve jobs according to SRPT. We thus reduce the problem to determining the release times of completed jobs. For the clearing, or transient system, where all jobs are present at time 0, we give a complete characterization of the optimal policy and show that it is fully determined by the cost-to-capacity ratio. With arrivals, the problem is much more complicated, and we can obtain only partial results.

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The virtual waiting-time and related processes

Advances in Applied Probability ◽

10.1017/s0001867800015883 ◽

1986 ◽

Vol 18 (02) ◽

pp. 558-573 ◽

Cited By ~ 6

Author(s):

D. R. Cox ◽

Valerie Isham

Keyword(s):

Waiting Time ◽

Arrival Rate ◽

Transient Behaviour ◽

Decay Properties ◽

State Dependent ◽

Continuous State Space ◽

Virtual Waiting Time ◽

Continuous State ◽

Waiting Time Process ◽

Properties Of Soil

The virtual waiting-time process of Takács is one of the simplest examples of a stochastic process with a continuous state space in continuous time in which jump transitions interrupt periods of deterministic decay. Properties of the process are reviewed, and the transient behaviour examined in detail. Several generalizations of the process are studied. These include two-sided jumps, periodically varying ‘arrival’ rate and the presence of a state-dependent decay rate; the last case is motivated by the properties of soil moisture in hydrology. Throughout, the emphasis is on the derivation of simple interpretable results.

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Teletraffic engineering for product-form circuit-switched networks

Advances in Applied Probability ◽

10.1017/s0001867800019935 ◽

1990 ◽

Vol 22 (03) ◽

pp. 657-675 ◽

Cited By ~ 1

Author(s):

Keith W. Ross ◽

Danny Tsang

Keyword(s):

Arrival Rate ◽

Product Form ◽

Long Distance ◽

Tree Networks ◽

Switched Networks ◽

Modeling Methodology ◽

State Dependent ◽

Point To Point ◽

Erlang Loss ◽

Access Link

We develop a performance modeling methodology for product-form circuit-switched networks. These networks allow for: arbitrary topology and link capacities; Poisson and finite population arrivals; multiple classes of calls, each class with a different route and bandwidth requirement; conference as well as point-to-point calls. The methodology is first applied to generalized tree networks, which consist of multiple access links feeding into a common link. Each access link may support multiple ‘long-distance' classes (requiring circuits only on the access link and on the common link) and multiple ‘local' classes (requiring circuits only on the access link). For generalized tree networks an efficient algorithm is given to determine the blocking probabilities. The methodology is then applied to hierarchical tree networks, where traffic is repeatedly merged in the direction of a root node. We also establish a ‘Norton' theorem for product-form circuit-switched networks. This theorem implies that for any given calling class, the entire network can be replaced by an Erlang loss system with a state-dependent arrival rate, without modifying the equilibrium probabilities for the particular calling class.

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Service Rate Optimization of Finite Population Queueing Model with State Dependent Arrival and Service Rates

Journal of the Institute of Engineering ◽

10.3126/jie.v13i1.20348 ◽

2018 ◽

Vol 13 (1) ◽

pp. 60-68

Author(s):

Sushil Ghimire ◽

Gyan Bahadur Thapa ◽

Ram Prasad Ghimire

Keyword(s):

Finite Population ◽

Arrival Rate ◽

Service Rate ◽

Queueing Model ◽

State Dependent ◽

Service Rates ◽

Rate Optimization

Providing service immediately after the arrival is rarely been used in practice. But there are some situations for which servers are more than the arrivals and no one has to wait to get served. In this model, arrival rate is

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