Monotonicity of the Loss Probabilities of Single Server Finite Queues with Respect to Convex Order of Arrival or Service Processes (original) (raw)
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Applied Mathematical Modelling, 2013
In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.
V.M.: Losses in M/GI/m/n queues
2007
The M/GI/m/n queueing system with m homogeneous servers and the finite number n of waiting spaces is studied. Let λ be the customers arrival rate, and let µ be the reciprocal of the expected service time of a customer. Under the assumption λ = mµ it is proved that the expected number of losses during a busy period is the same value for all n ≥ 1, while in the particular case of the Markovian system M/M/m/n the expected number of losses during a busy period is m m m! for all n ≥ 0. Under the additional assumption that the probability distribution function of a service time belongs to the class NBU or NWU, the paper establishes simple inequalities for those expected numbers of losses in M/GI/m/n queueing systems.
Minimizing the Loss Probability in M/M/2/1 Queueing System with Ordered Entry
American Journal of Operations Research
This study analyzed the M/M/2/1 queueing model with queue of length one (waiting room of capacity just one), heterogeneous servers and ordered entry using the method of semi-Markov process. The customers who arrive in the system enter the free server; if the two servers are free, the customers enter the first server. If the two servers are busy, just one customer can wait at the waiting room. If the two servers are busy and the waiting room has a customer, the following customers will leave the system without receiving any service. Such a customer is called LOST COSTOMER. The probability of lost customers in the queueing system under examination was computed. Furthermore, by using inequality () e as f s − ≥ obtained from Jensen's inequality, it was shown that the loss probability was minimum when inter-arrival times fit deterministic distribution [1] [2].
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2008
A non-classical single-server queueing system with non-homogeneous customers having some random space requirement (capacity, volume) can be used as a model of a wide class of computer and communicating systems. We assume that the total customers capacity in the system is limited by some constant value V > 0 that is called the value of memory capacity of the system. Service time of a customer generally depends on his capacity. For such systems we determine some estimators of stationary loss characteristics and compare the analitical results with ones obtained by simulation.
Computers & Industrial Engineering, 2006
The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.
On the subexponential properties in stationary single-server queues: a Palm-martingale approach
Advances in Applied Probability, 2004
This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale ca...
For the M/G/1/m queue with the forced vacations in work after n customers served in succession probabilities of states of limiting steady-state process are found. 1. Introduction. In case of arbitraryly distributed service time and exponentially distributed interarrival time existence of limiting steady-state process for the single-server queueing system with unlimited queue is proved by a method of embedded Markov chains [ 1, p. 98 ]. For M/G/1/m queue probabilities of states of limiting steady-state process are found in [ 2, p. 235 ]. Below we study M/G/1/m queue which feature consists that after ï customers served in succession the forced vacations in work of system occur. Necessity of such pause can be caused by technological or other reasons. For example, the technical device which consistently carries out homogeneous operations, at designing can be calculated on the limited number of operations which are carried out without interruption between operations.
Communications in Mathematics
This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing servic...