A Markov chain approach to periodic queues

1987 ◽  
Vol 24 (1) ◽  
pp. 215-225 ◽  
Author(s):  
Søren Asmussen ◽  
Hermann Thorisson
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Suitable Condition ◽  
Time Dependent ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Markov Chain Approach ◽  
Periodic Queues ◽  
Queue Lengths ◽  
Dependent Processes

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.

A Markov chain approach to periodic queues

1987 ◽  
Vol 24 (01) ◽  
pp. 215-225 ◽  
Author(s):  
Søren Asmussen ◽  
Hermann Thorisson
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Suitable Condition ◽  
Time Dependent ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Markov Chain Approach ◽  
Periodic Queues ◽  
Queue Lengths ◽  
Dependent Processes

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n } and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.

FORWARD-RATE VOLATILITIES AND THE SWAPTION MATRIX: WHY NEITHER TIME-HOMOGENEITY NOR TIME-DEPENDENCE ARE ENOUGH

2006 ◽  
Vol 09 (05) ◽  
pp. 705-746 ◽  
Author(s):  
RICCARDO REBONATO
Keyword(s):  
Markov Chain ◽  
Market Volatility ◽  
Time Dependent ◽  
Forward Rate ◽  
Systematic Analysis ◽  
Option Markets ◽  
Markov Chain Approach ◽  
Major Market ◽  
Volatility Model

This work presents the first systematic analysis of the whole swaption matrix by fitting a parsimonious, nonlinear, financially-inspired volatility model to market data. The study uses several years of data spanning period of major market volatility. We find that the quality of the fits is good (on average of the same magnitude as the bid-offer spread), and better when a displaced-diffusion approach is chosen, but some systematic shortcomings are observed and discussed. The analysis suggests that a two-regime Markov chain approach may be more successful and better financially motivated. More generally, the present study highlights the shortcomings of purely time-dependent or time-homogenous approaches. These findings should be applicable to other option markets as well. Finally, we find that the present (nonlinear) model vastly outperforms PCA-based approaches when in comes to predicting moves in implied volatilities.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

1995 ◽  
Vol 27 (02) ◽  
pp. 532-566 ◽  
Author(s):  
John S. Sadowsky ◽  
Wojciech Szpankowski
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Renewal Theory ◽  
Batch Size ◽  
Interarrival Time ◽  
Recurrent Markov Chain ◽  
Queue Lengths ◽  
Exponential Change ◽  
Stationary Probabilities

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (4) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S′−x)+ ≦ E(S″−x)+ (all x >0), are stochastically ordered as W′≦dW″. The weaker conclusion, that E(W′−x)+ ≦ E(W″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T′)+ ≦ E(x−T″)+ (all x). A sufficient condition for wk≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

1995 ◽  
Vol 27 (2) ◽  
pp. 532-566 ◽  
Author(s):  
John S. Sadowsky ◽  
Wojciech Szpankowski
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Renewal Theory ◽  
Batch Size ◽  
Interarrival Time ◽  
Recurrent Markov Chain ◽  
Queue Lengths ◽  
Exponential Change ◽  
Stationary Probabilities

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (04) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S ′−x)+ ≦ E(S ″−x)+ (all x >0), are stochastically ordered as W ′≦d W ″. The weaker conclusion, that E(W ′−x)+ ≦ E(W ″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T ′)+ ≦ E(x−T ″)+ (all x). A sufficient condition for wk ≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

Estimation of Interregional Trade Flows: A Markov Chain Approach

1986 ◽  
Vol 18 (1) ◽  
pp. 123-132 ◽  
Author(s):  
I Weksler ◽  
D Freeman ◽  
G Alperovich
Keyword(s):  
Markov Chain ◽  
Trade Flows ◽  
Interregional Trade ◽  
Markov Chain Approach
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