Title
LongRange Dependence of Markov Chains: Theory and Examples,Used
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Longrange dependence in discrete and continuous time Markov chains over a countable state space is defined via embedded renewal processes brought about by visits to a fixed state. In the discrete time chain, solidarity properties are obtained and longrange dependence of functionals are examined. On the other hand, the study of LRD of continuous time chains is defined via the number of visits in a given time interval. Longrange dependence of Markov chains over a noncountable state space is also carried out through positive Harris chains. Examples of these chains are presented, with particular attention given to longrange dependent Markov chains in singleserver queues, namely, the waiting times of GI/G/1 queues and queue lengths at departure epochs in M/G/1 queues. The presence of longrange dependence in these processes is dependent on the moment index of the lifetime distribution of the service times. These processes of waiting times and queue sizes are also examined in a range of M/P/2 queues via simulation (here, $P$ denotes a Pareto distribution).
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Shipping & Returns
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We ship your order within 2–3 business days for USA deliveries and 5–8 business days for international shipments. Once your package has been dispatched from our warehouse, you'll receive an email confirmation with a tracking number, allowing you to track the status of your delivery.
Returns
To facilitate a smooth return process, a Return Authorization (RA) Number is required for all returns. Returns without a valid RA number will be declined and may incur additional fees. You can request an RA number within 15 days of the original delivery date. For more details, please refer to our Return & Refund Policy page.
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Warranty
We provide a 2-year limited warranty, from the date of purchase for all our products.
If you believe you have received a defective product, or are experiencing any problems with your product, please contact us.
This warranty strictly does not cover damages that arose from negligence, misuse, wear and tear, or not in accordance with product instructions (dropping the product, etc.).
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Your payment information is processed securely. We do not store credit card details nor have access to your credit card information.
We accept payments with :
Visa, MasterCard, American Express, Paypal, Shopify Payments, Shop Pay and more.