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### On the k-server conjecture

1994 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing - STOC '94
Manasse et al.  conjectured that the competitive ratio for the k-server problem is exactly k (it is trivially at least k); previously the best-known upper bound was exponential in k.  ...  We prove that the work function algorithm for the k-server problem has a competitive ratio at most 2k -1.  ...  In Manasse et al.  , it was shown that no on-line algorithm can have a competitive ratio less that k and the following conjecture was posed: CONJECTURE 1.1 (THE k-SERVER CONJECTURE), For elley metric  ...

### On the k-server conjecture

1995 Journal of the ACM
Manasse et al.  conjectured that the competitive ratio for the k-server problem is exactly k (it is trivially at least k); previously the best-known upper bound was exponential in k.  ...  We prove that the work function algorithm for the k-server problem has a competitive ratio at most 2k -1.  ...  In Manasse et al.  , it was shown that no on-line algorithm can have a competitive ratio less that k and the following conjecture was posed: CONJECTURE 1.1 (THE k-SERVER CONJECTURE), For elley metric  ...

### On the Monotonicity of Work Function in k-Server Conjecture [article]

Ming-Zhe Chen
2008 arXiv   pre-print
This paper presents a mistake in work function algorithm of k-server conjecture. That is, the monotonicity of the work function is not always true.  ...  Introduction The k-server conjecture has not been proved. A lot of literature deal with the k-server conjecture     and references therein.  ...  We still think WFA would be k-competitive. d(a, b) = 1, d(a, c) = 7, d(a, d) = 5, d(a, e) = 8, d(b, c) = 4, d(b, d) = 2, d(b, e) = 10, d(c, d) = 3, d(c, e) = 9, d(d, e) = 6 Consider 3-servers on this graph  ...

### The -server problem

Elias Koutsoupias
2009 Computer Science Review
The k-server conjecture, which was posed more that two decades ago when the problem was first studied within the competitive analysis framework, is still open and has been a major driving force for the  ...  This article surveys some major results for the k-server. * University of Athens, elias@di.uoa.gr.  ...  This is supported by all results on the problem in the last two decades and it is exactly what the k-server conjecture states: Conjecture 1 (The k-server conjecture).  ...

### Page 5195 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews
However, we provide credence to the conjectures by establishing that for balanced sys- tems, the conjectured asymptotic loss of the HW policy is a lower bound on the asymptotic loss of all policies, while  ...  Summary: “In this paper we analyze a discrete-time single server queue where the service time equals one slot.  ...

### A lower bound for the Erlang C formula in the Halfin–Whitt regime

A. J. E. M. Janssen, Johan S. H. van Leeuwaarden, Bert Zwart
2011 Queueing systems
Mathematics Subject Classification (2000) 60K25 One of the classical models of queueing theory is the M/M/s queue or Erlang delay model. This model has s homogeneous servers working in parallel.  ...  This probability is known as the Erlang C formula, given by (with ρ = a/s) C(s, a) = a s s!(1−ρ) s−1 k=0 a k /k! + a s s!  ...  Acknowledgements We thank an anonymous referee for subjecting Conjecture 1 to close scrutiny and for suggesting Conjecture 2.  ...

### Introduction to special issue on open problems

2011 Queueing systems
In addition, if the open problem is in the form of a conjecture, there is some motivation why the author/authors thinks/think the conjecture is true.  ...  The open problems thus provide the readership with a state of the art, and will hopefully trigger new efforts, new collaborations, and new approaches.  ...  First Gupta and Osogami postulate a conjecture regarding upper and lower bounds on the mean waiting time for the M/G/K queue, analogously to the classical Markov-Krein theorem.  ...

### On the competitive ratio of the work function algorithm for the k-server problem

Yair Bartal, Elias Koutsoupias
2004 Theoretical Computer Science
The k-server problem is one of the most fundamental online problems. The problem is to schedule k mobile servers to visit a sequence of points in a metric space with minimum total mileage.  ...  The k-server conjecture of Manasse, McGeogh, and Sleator states that there exists a k-competitive online algorithm. The conjecture has been open for over 15 years.  ...  The notorious k-server conjecture states that there is an online algorithm that has competitive ratio k on any metric space.  ...

### A Survey of Stability Results for Redundancy Systems [article]

Elene Anton
2021 arXiv   pre-print
Redundancy mechanisms consist in sending several copies of a same job to a subset of servers. It constitutes one of the most promising ways to exploit diversity in multiservers applications.  ...  We also point out open questions and conjectures.  ...  Ayesta has received funding from the Department of Education of the Basque Government through the Consolidated Research Group MATHMODE (IT1294-19) .  ...

### Towards the Capacity of Private Information Retrieval from Coded and Colluding Servers [article]

Lukas Holzbaur, Ragnar Freij-Hollanti, Jie Li, Camilla Hollanti
2021 arXiv   pre-print
This positively settles the capacity conjectures stated by Freij-Hollanti et al. and Tajeddine et al. in the presented cases.  ...  servers for the recently introduced concept of matched randomness.  ...  ACKNOWLEDGMENT The authors would like to thank Prof. Syed Jafar, Prof. Chao Tian, and Dr. Qiwen Wang for helpful discussions.  ...

### The Delayed k-Server Problem [chapter]

Wolfgang W. Bein, Kazuo Iwama, Lawrence L. Larmore, John Noga
2005 Lecture Notes in Computer Science
We show that the delayed k-server problem is equivalent to the (k − 1)-server problem in the uniform case, but not in general.  ...  In this problem, once a server moves to serve a request, it must wait for one round to move again, but could serve a repeated request to the same point.  ...  Acknowledgement Wolfgang Bein thanks Kazuo Iwama for the generous support he received while he visited Kyoto University to work on this project during December 2004 and January 2005.  ...

### A General Decomposition Theorem for the k-Server Problem

Steven S. Seiden
2002 Information and Computation
The first general decomposition theorem for the k-server problem is presented.  ...  Whereas all previous theorems are for the case of a finite metric with k + 1 points, the theorem given here allows an arbitrary number of points in the underlying metric space.  ...  Central to the k-server problem is the k-server conjecture, which proposes that there is a deterministic k-server algorithm which is k-competitive for all metrics.  ...

### A general private information retrieval scheme for MDS coded databases with colluding servers [article]

Yiwei Zhang, Gennian Ge
2017 arXiv   pre-print
We build a general private information retrieval scheme for MDS coded databases with colluding servers. Our scheme achieves the rate (1+R+R^2+...+R^M-1), where R=1-N-T K/N K.  ...  PIR capacity is the maximal number of bits privately retrieved per one bit of downloaded bit. The capacity has been fully solved for some degenerating cases.  ...  For the non-degenerating cases, one can see that our rate is strictly smaller than the conjectured capacity in (2) .  ...

### Design Heuristic for Parallel Many Server Systems under FCFS-ALIS [article]

Ivo Adan, Marko Boon, Gideon Weiss
2018 arXiv   pre-print
We consider the service policy that always assigns servers to the first, longest waiting compatible customer, and that always assigns customers to the longest idle compatible server if on arrival, multiple  ...  In particular, the calculation of matching rates, the fraction of services of customer-server type, is intractable.  ...  Khinchine  ), which indicates that our conjecture on server availability may in fact be true.  ...

### Stability of multi-server polling system with server limits

Nelson Antunes, Christine Fricker, James Roberts
2011 Queueing systems
We postulate a conjecture for the stability condition in this case that is supported by our simulation results.  ...  We consider a multi-server polling system with server limits, that is the number of servers that can attend a queue simultaneously is limited.  ...  The stability condition of a multi-server polling system with a K-limited service policy (i.e., on each visit, the server takes K customers or empties the queue if number waiting is less than K) 1 was  ...
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