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In [[queueing theory]], a discipline within the mathematical [[theory of probability]], '''mean value analysis (MVA)''' is a recursive technique for computing [[expected value|expected]] queue lengths, waiting time at queueing nodes and throughput in equilibrium for a closed separable system of queues. The first approximate techniques were published indepdenently by Schweitzer<ref name="Schweitzer" /> and Bard,<ref>{{cite journal | title = Some Extensions to Multiclass Queueing Network Analysis | first = Yonathan | last = Bard | year = 1979 | journal = Proceedings of the Third International Symposium on Modelling and Performance Evaluation of Computer Systems: Performance of Computer Systems | pages = 51–62 | isbn = 0-444-85332-4 | publisher = North-Holland Publishing Co.}}</ref><ref>{{cite doi|10.1007/978-1-4419-6472-4_13}}</ref> followed later by an exact version by Lavenberg and Reiser published in 1980.<ref>{{cite doi|10.1145/322186.322195}}</ref><ref>{{cite doi|10.1007/3-540-46506-5_22}}</ref>
Questioning what to do when you have been in an accident, a slip and fall, or a workplace injury? If you have currently spoken with in insurance provider for your insurer, it might be time to contemplate a private injury attorney.<br><br>The danger in running right out and hiring a personal injury attorney immediately soon after injury is that you will have to pay for their solutions out of what ever payout you ultimately get.  So, it is typically wise to very first speak with the relevant insurance provider and only then turn to legal options. Even though speaking to the insurance firm will in many scenarios resolve the problem completely, there may possibly be some conditions in which the insurance coverage firm either denies your claim entirely or decides to compensate you in a matter that is from your point of view insufficient. In such situations, the premium that you pay out to a individual injury attorney will come back to you in settlement damages several occasions more than.<br><br>Now, after you are convinced that you want a individual injury lawyer, you have to put forth the effort of first locating the appropriate particular person for your case. Right here are a couple of ideas on picking the proper lawyer for you:<br><br>There are a quantity of online databases of neighborhood and regional personal injury attorneys.  Whilst most of these are spend per listing or free submission, some truly do provide reviewing solutions.  Even right here, nonetheless, be wary of placing too much stock in a internet sites recommendation as this information can be simply manipulated.<br><br>Employing this on the web database and your neighborhood yellow pages as a sort of basic list, it then becomes crucial to narrow this list by looking at the credentials of a specific lawyer. For another perspective, people might claim to view at: [http://www.minghuigongye.com/showthread.php?tid=4013 A Criminal Defense Lawyer Will Help You Prevent A Misdemeanor Demand].  Probably the greatest way to do this is to call your regional legal aid clinic, which is totally free. Although these individuals are not in the organization of supplying suggestions, the bar in a specific city for individual injury typically consists of about fifty to one particular hundred attorneys, so amongst attorneys word gets around pretty rapidly as to who is excellent and who is not.<br><br>An additional excellent resource is your city bar association. Your city bar will hold more basic data about who has not been sanctioned by courts for malpractice and typically keeps a quick list of advised attorneys. But once again, take this list with a grain of salt as there are manipulations going on behind the scenes here as well. Frequently a far better tack is just to have a list of 3 or four attorneys you are thinking about and then get in touch with the bar association to hear their thoughts on every one particular.<br><br>Referrals can be good if you know an individual in the legal neighborhood. Browse here at the link [http://www.yjzdc.com/showthread.php?tid=33013 A Criminal Defense Lawyer Can Help You Prevent A Misdemeanor Cost] to learn why to allow for it.  But if not, the ideal you will get out of a referral is a sort of he/she isnt incompetent. Since lets face it, most clientele have no notion regardless of whether the settlement they received was in actuality the very best they could have.<br><br>The final test should constantly be to meet with the lawyer.  In such a meeting you can talk about the payment strategy (contingency or flat-charge), but far more importantly you want to attempt to assess if this person appears competent. Note issues like, what law school he/she attended?  Did they pass the bar on their initial try? How extended have they been practicing individual injury in your state? Will they let you speak with some of their most recent consumers?  All of these elements taken with each other ought to give you a common impression of regardless of whether your individual injury attorney is competent, if not great..<br><br>If you adored this article and you also would like to acquire more info relating to [http://storify.com/direfulcus283 health and fitness] i implore you to visit the webpage.
 
It is based on the [[arrival theorem]], which states that when one customer in an ''M''-customer closed system arrives at a service facility he/she observes the rest of the system to be in the equilibrium state for a system with ''M''&nbsp;&minus;&nbsp;1 customers.
 
==Problem setup==
Consider a closed queueing network of ''K'' [[M/M/1 queue]]s, with ''M'' customers circulating in the system. To compute the mean queue length and waiting time at each of the nodes and throughput of the system we use an iterative algorithm starting with a network with 0 customers.
 
Write ''μ''<sub>''i''</sub> for the service rate at node ''i'' and ''P'' for the customer routing matrix where element ''p''<sub>''ij''</sub> denotes the probability that a customer finishing service at node ''i'' moves to node ''j'' for service. To use the algorithm we first compute the visit ratio row vector '''v''', a vector such that '''v'''&nbsp;=&nbsp;'''v'''&nbsp;P.
 
Now write ''L''<sub>''i''</sub>(''n'') for the mean number of customer at queue ''i'' when there are a total of ''n'' customers in the system (this includes the job currently being served at queue ''i'') and ''W''<sub>''j''</sub>(''n'') for the mean time spent by a customer in queue ''i'' when there are a total of ''n'' customers in the system. Denote the throughput of a system with ''m'' customers by ''λ''<sub>''m''</sub>.
 
==Algorithm==
The algorithm<ref>{{cite book |first=Sanjay K.|last=Bose|title=An introduction to queueing systems|publisher=Springer|year=2001|isbn=0-306-46734-8|url=http://books.google.com/books?id=39-jISti_zkC|page=174}}</ref> starts with an empty network (zero customers), then increases the number of customers by 1 at each iteration until there are the required number (''M'') of customers in the system.
 
To initialise, set ''L''<sub>''k''</sub>(0)&nbsp;=&nbsp;0 for ''k''&nbsp;=&nbsp;1,...,''K''. (This sets the average queue length in a system with no customers to zero at all nodes.)
 
Repeat for ''m''&nbsp;=&nbsp;1,...,''M'':
 
:1. For ''k''&nbsp;=&nbsp;1, ..., ''K'' compute the waiting time at each node using the arrival theorem
:::<math>W_k(m) = \frac{L_k\left(m-1\right)+1}{\mu_k}.</math>
:2. Then compute the system throughput using Little's law
:::<math>\lambda_m=\frac{m}{\sum_{k=1}^K W_k(m) v_k}.</math>
:3. Finally, use little's law applied to each queue to compute the mean queue lengths for ''k''&nbsp;=&nbsp;1, ..., ''K''
:::<math>L_k(m)=v_k \lambda_m W_k(m).</math>
 
End repeat.
 
==Schweitzer's approximation==
 
Schweitzer's approximation estimates the average number of jobs at node ''k'' to be<ref name="Schweitzer">{{cite doi|10.1007/BFb0013865}}</ref><ref>{{cite journal | last = Schweitzer | first = Paul | title = Approximate analysis of multiclass closed networks of queues | journal = Proceedings of International Conference on Stochastic Control and Optimization | year = 1979}}</ref>
 
:<math>L_k(m-1) \approx \frac{m-1}{m} L_k(m)</math>
 
which from the above formulas yields [[fixed-point iteration|fixed-point relationships]] which can be solved numerically. This iterative approach is typically faster than the recursive approach of MVA.<ref>{{cite doi|10.2200/S00282ED1V01Y201005CSL002}}</ref>{{rp|38}}
 
===Pseudo-code===
 
<tt>
set ''L''<sub>''k''</sub>(''m'') = ''M''/''K''
 
repeat until convergence:
 
::<math>\lambda_m = \frac{m}{\sum_{k=1}^K \frac{\frac{m-1}{m}L_k(m) + 1}{\mu_k} v_k}</math>
::<math>L_k(m) = v_k \lambda_m \frac{\frac{m-1}{m}L_k(m) + 1}{\mu_k}</math>
 
</tt>
 
==Multiclass networks==
 
For networks with a single customer class the MVA algorithm is very fast and time taken grows linearly with the number of customers and number of queues. However, the number of multiplications and additions required for MVA grows exponentially with the number of customer classes. Practically, the algorithm works for 3-4 customer classes.<ref name="casale">{{cite doi|10.1016/j.peva.2010.12.009}}</ref> The ''method of moments'' is an exact method which required log-quadratic time and can solve models with up to 10 classes of customers.<ref name="casale" /> Approximate algorithms have also been proposed with lower complexity.<ref>{{cite doi|10.1145/79147.214074}}</ref>
 
==Extensions==
 
The mean value analysis algorithm has been applied to a class of [[PEPA]] models describing [[queueing network]]s and the performance of a [[key distribution center]].<ref>{{cite doi|10.1093/comjnl/bxq064}}</ref>
 
==Software==
 
*[http://jmt.sourceforge.net/JMVA.html JMVA], a tool written in [[Java (programming language)|Java]] which implements MVA.<ref>{{cite doi|10.1145/1530873.1530877}}</ref>
 
==External links==
*[http://www.netlab.tkk.fi/opetus/s383143/kalvot/E_qnets.pdf J. Virtamo: Queuing networks]. Handout from Helsinki Tech gives good overview of Jackson's Theorem and MVA.
*[http://www.cs.utexas.edu/users/lam/Vita/Jpapers/Lam83.pdf Simon Lam: A simple derivation of the MVA algorithm]. Shows relationship between [[Buzen's algorithm]] and MVA.
 
==References==
{{reflist}}
 
{{Queueing theory}}
 
[[Category:Stochastic processes]]
[[Category:Queueing theory]]

Latest revision as of 08:03, 9 January 2015

Questioning what to do when you have been in an accident, a slip and fall, or a workplace injury? If you have currently spoken with in insurance provider for your insurer, it might be time to contemplate a private injury attorney.

The danger in running right out and hiring a personal injury attorney immediately soon after injury is that you will have to pay for their solutions out of what ever payout you ultimately get. So, it is typically wise to very first speak with the relevant insurance provider and only then turn to legal options. Even though speaking to the insurance firm will in many scenarios resolve the problem completely, there may possibly be some conditions in which the insurance coverage firm either denies your claim entirely or decides to compensate you in a matter that is from your point of view insufficient. In such situations, the premium that you pay out to a individual injury attorney will come back to you in settlement damages several occasions more than.

Now, after you are convinced that you want a individual injury lawyer, you have to put forth the effort of first locating the appropriate particular person for your case. Right here are a couple of ideas on picking the proper lawyer for you:

There are a quantity of online databases of neighborhood and regional personal injury attorneys. Whilst most of these are spend per listing or free submission, some truly do provide reviewing solutions. Even right here, nonetheless, be wary of placing too much stock in a internet sites recommendation as this information can be simply manipulated.

Employing this on the web database and your neighborhood yellow pages as a sort of basic list, it then becomes crucial to narrow this list by looking at the credentials of a specific lawyer. For another perspective, people might claim to view at: A Criminal Defense Lawyer Will Help You Prevent A Misdemeanor Demand. Probably the greatest way to do this is to call your regional legal aid clinic, which is totally free. Although these individuals are not in the organization of supplying suggestions, the bar in a specific city for individual injury typically consists of about fifty to one particular hundred attorneys, so amongst attorneys word gets around pretty rapidly as to who is excellent and who is not.

An additional excellent resource is your city bar association. Your city bar will hold more basic data about who has not been sanctioned by courts for malpractice and typically keeps a quick list of advised attorneys. But once again, take this list with a grain of salt as there are manipulations going on behind the scenes here as well. Frequently a far better tack is just to have a list of 3 or four attorneys you are thinking about and then get in touch with the bar association to hear their thoughts on every one particular.

Referrals can be good if you know an individual in the legal neighborhood. Browse here at the link A Criminal Defense Lawyer Can Help You Prevent A Misdemeanor Cost to learn why to allow for it. But if not, the ideal you will get out of a referral is a sort of he/she isnt incompetent. Since lets face it, most clientele have no notion regardless of whether the settlement they received was in actuality the very best they could have.

The final test should constantly be to meet with the lawyer. In such a meeting you can talk about the payment strategy (contingency or flat-charge), but far more importantly you want to attempt to assess if this person appears competent. Note issues like, what law school he/she attended? Did they pass the bar on their initial try? How extended have they been practicing individual injury in your state? Will they let you speak with some of their most recent consumers? All of these elements taken with each other ought to give you a common impression of regardless of whether your individual injury attorney is competent, if not great..

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