Approximate Solution Approach and Performability Evaluation of Large Scale Beowulf Clusters

Yonal Kirsal; Yoney Kirsal Ever

arxiv: 1907.08044 · v1 · pith:DFLJCTPEnew · submitted 2019-07-18 · 💻 cs.PF

Approximate Solution Approach and Performability Evaluation of Large Scale Beowulf Clusters

Yonal Kirsal , Yoney Kirsal Ever This is my paper

Pith reviewed 2026-05-24 19:38 UTC · model grok-4.3

classification 💻 cs.PF

keywords Beowulf clustersperformability evaluationapproximate solutionanalytical modelQoSavailabilitylarge-scale systemsserver farms

0 comments

The pith

An approximate analytical method evaluates performability and QoS for large Beowulf clusters where exact models fail.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Beowulf clusters deliver high computational power for scientific computing but struggle with availability at large scales due to faults and varied workloads. The paper constructs an analytical model for quality of service in these systems and introduces a generic approximate solution technique because exact modeling cannot scale with node count or request diversity. The method supplies flexibility to compute metrics such as availability. Accuracy is checked by matching outputs against discrete event simulations. This supplies a practical route to predict behavior in widely deployed server farms that handle huge data volumes.

Core claim

The paper claims that a generic and flexible approximate solution approach can be developed to handle large numbers of nodes for performability evaluation of Beowulf clusters, together with an analytical model for QoS that accounts for availability issues, with the results validated against discrete event simulations to demonstrate efficacy and accuracy.

What carries the argument

The approximate solution approach applied to an analytical performability model of Beowulf clusters.

If this is right

QoS measurements become evaluable for large-scale Beowulf clusters and server farms.
Availability issues can be incorporated into analytical predictions of system behavior.
The approach supplies flexibility to assess different performance and fault scenarios.
Validation against simulations confirms the method's practical utility for high-performance computing systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same approximation strategy could extend to other distributed computing platforms beyond Beowulf clusters.
Adding more detailed workload or failure patterns would test the method's robustness on real deployments.
Implementation in monitoring software might allow ongoing QoS estimation during operation.

Load-bearing premise

Exact modeling of such clusters is not feasible due to the nature of the large scale nodes and the diversity of user requests.

What would settle it

A direct comparison in which the approximate model's QoS predictions deviate substantially from results of discrete event simulations on a Beowulf cluster with thousands of nodes would falsify the claimed accuracy.

read the original abstract

Beowulf clusters are very popular and deployed worldwide in support of scientific computing, because of the high computational power and performance. However, they also pose several challenges, and yet they need to provide high availability. The practical large-scale Beowulf clusters result in unpredictable, fault-tolerant, often detrimental outcomes. Successful development of high performance in storing and processing huge amounts of data in large-scale clusters necessitates accurate quality of service (QoS) evaluation. This leads to develop as well as design, analytical models to understand and predict of complex system behaviour in order to ensure availability of large-scale systems. Exact modelling of such clusters is not feasible due to the nature of the large scale nodes and the diversity of user requests. An analytical model for QoS of large-scale server farms and solution approaches are necessary. In this paper, analytical modelling of large-scale Beowulf clusters is considered together with availability issues. A generic and flexible approximate solution approach is developed to handle large number of nodes for performability evaluation. The proposed analytical model and the approximate solution approach provide flexibility to evaluate the QoS measurements for such systems. In order to show the efficacy and the accuracy of the proposed approach, the results obtained from the analytical model are validated with the results obtained from the discrete event simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a standard queueing-based approximation for performability in large Beowulf clusters and claims simulation validation, but the validation details needed to support the large-N claim are missing from the abstract.

read the letter

The main takeaway is that the authors build an approximate analytical model for QoS and availability in large-scale Beowulf clusters and check it against discrete-event simulation. They argue exact Markov models become intractable with many nodes and varied requests, so the approximation is meant to fill that gap and give flexible QoS numbers. That framing is reasonable for an applied modeling paper. The work does a straightforward job of combining performance and availability into one model and showing how the approximation can be used for server-farm style systems. Validation against simulation is the correct check to run. The approach appears to reuse established queueing and availability techniques rather than derive something new from first principles, which is fine if the goal is a practical tool rather than a methodological advance. The soft spot is exactly the one in the stress-test note. The abstract states that the analytical results match simulation but gives no cluster sizes, no error values, and no indication that the tested N is large enough that an exact chain would be impossible. If the simulations stay in the modest-N range where exact solutions are still feasible, the accuracy claim for the regime where the approximation is actually required stays unproven. That is a real gap in the evidence presented. The paper is aimed at researchers who model cluster performance and need workable QoS estimates under failures. A reader already working in queueing models for distributed systems could extract the model structure and try the approximation on their own cases. It shows honest engagement with the practical constraints of scale and availability, with no obvious internal contradictions in the stated approach. I would send it to peer review so referees can see the full equations, the exact simulation parameters, and whether the large-N accuracy holds up.

Referee Report

2 major / 1 minor

Summary. The paper develops an analytical model for the performability and QoS of large-scale Beowulf clusters together with a generic approximate solution approach intended to handle large node counts where exact modeling is claimed to be infeasible; the approach is asserted to be flexible for evaluating availability and performance metrics, with accuracy demonstrated by comparison to discrete-event simulation results.

Significance. If the approximation is shown to remain accurate precisely in the large-N regime where both exact Markov analysis and full simulation become intractable, the work would supply a practical tool for QoS prediction and design of fault-tolerant high-performance computing clusters. The manuscript does not yet supply the quantitative evidence needed to establish this regime-specific accuracy.

major comments (2)

[Abstract] Abstract: the central claim that the approximate solution enables performability evaluation 'for large number of nodes' rests on validation against discrete-event simulations, yet the abstract (and, on the information given, the manuscript) supplies no details on the node counts N used in those simulations, the error metrics employed, or whether the simulated instances reach the scale at which exact analysis is intractable. This information is load-bearing for the claim that the method is useful precisely where exact modeling fails.
[Abstract] Abstract: the statement that 'exact modelling of such clusters is not feasible due to the nature of the large scale nodes' is presented as a premise without supporting argument or reference to state-space explosion results; if the subsequent approximation is derived under this premise, the lack of a concrete demonstration that the target N exceeds the feasible exact-model range weakens the justification for the approximate approach.

minor comments (1)

[Abstract] Abstract contains several grammatical issues ('predict of complex system behaviour', 'understand and predict of') that should be corrected for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and commit to revisions that strengthen the manuscript's claims regarding the large-N regime.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the approximate solution enables performability evaluation 'for large number of nodes' rests on validation against discrete-event simulations, yet the abstract (and, on the information given, the manuscript) supplies no details on the node counts N used in those simulations, the error metrics employed, or whether the simulated instances reach the scale at which exact analysis is intractable. This information is load-bearing for the claim that the method is useful precisely where exact modeling fails.

Authors: We agree that the abstract must explicitly report the validation scale, error metrics, and intractability threshold to support the central claim. In the revised version we will expand the abstract to state the node counts employed in the discrete-event simulations, the quantitative error metrics used for comparison, and the point at which exact Markov analysis becomes infeasible. These additions will directly address the load-bearing information identified by the referee. revision: yes
Referee: [Abstract] Abstract: the statement that 'exact modelling of such clusters is not feasible due to the nature of the large scale nodes' is presented as a premise without supporting argument or reference to state-space explosion results; if the subsequent approximation is derived under this premise, the lack of a concrete demonstration that the target N exceeds the feasible exact-model range weakens the justification for the approximate approach.

Authors: We accept that the abstract presents the infeasibility premise without elaboration. We will revise the manuscript by adding a concise supporting clause (with a reference to the state-space explosion literature) either in the abstract or the opening of the introduction, together with a brief indication of the N values at which exact analysis ceases to be practical. This will provide the concrete demonstration requested. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper presents an analytical model for performability/QoS of large Beowulf clusters together with an approximate solution method, validated by comparison to independent discrete-event simulation results. No equations, parameter-fitting steps, or citations are shown in the provided text that would reduce any claimed prediction or uniqueness result to a definitional input or self-referential premise by construction. The validation is described as external confirmation rather than an internal re-derivation, leaving the core modeling steps self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate free parameters, axioms, or invented entities. The central claim rests on the premise that an approximate analytical model can usefully represent large-scale cluster behavior for QoS purposes.

pith-pipeline@v0.9.0 · 5755 in / 1053 out tokens · 22395 ms · 2026-05-24T19:38:33.199634+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Hwang, K., Geoﬀrey, C., Fox, J. J. Dongarra (2012). Distributed and Cloud Computing: From Parallel Processing to the Internet of Things. Morgan Kaufmann

work page 2012
[2]

Ngxande, M., Moorosi, N. (2014). Development of Beowulf cluster to perform large datasets simulations in educational institutions. Int. J. Computer App., 99, 29-35. Title Suppressed Due to Excessive Length 27

work page 2014
[3]

Ever, E., Gemikonakli, O., Chakka, R. (2006). A Mathematical Model for Highly Avail- able Clusters with One Head and Several Identical Computing Nodes. In Proceedings of the 9th International Conference on Computer Modelling and Simulation. 32-37

work page 2006
[4]

and Plourde, J

Pijanowski, B.C., Tayyebi, A., Doucette, J., Pekin, B.K., Braun, D. and Plourde, J. (2014). A big data urban growth simulation at a national scale: conﬁguring the GIS and neural network based land transformation model to run in a high performance computing (HPC) environment. Environmental Modelling and Software. 51. 250-268

work page 2014
[5]

Song, A., Wang, W., and Luo, J. (2014). Stochastic modeling of dynamic power manage- ment policies in server farms with setup times and server failures. International Journal of Communication Systems. 27. 680-703

work page 2014
[6]

Ekanayake, J., and Geoﬀrey, F. (2010). High performance parallel computing with clouds and cloud technologies. Cloud Computing. Springer Berlin Heidelberg. 20-38

work page 2010
[7]

Ever, E., Gemikonakli, O., Chakka, R. (2006). A Mathematical Model for Performability of Beowulf Clusters. In Proceedings of 39th Annual Symposium on Network Simulation. 118-126

work page 2006
[8]

B., Sterling, T., Savarese, D., Dorband, J., E., Ranawake, U., A., Packer, and Charles., P

Donald, J. B., Sterling, T., Savarese, D., Dorband, J., E., Ranawake, U., A., Packer, and Charles., P. V. (1995). Beowulf: A parallel workstation for scientiﬁc computation. In Pro- ceedings International Conference on Parallel Processing

work page 1995
[9]

and Vos, D

Adams, J. and Vos, D. (2002). Small-College Supercomputing: Building A Beowulf Cluster At A Comprehensive College. In Proceedings of 33rd Technical symposium on computer science education. Cincinnati. Kentucky. 411-415

work page 2002
[10]

Qin, C., Z. (2014). A strategy for raster-based geocomputation under diﬀerent parallel computing platforms. International Journal of Geographical Information Science. 28. 11. 2127-2144

work page 2014
[11]

Boukerche, A., Shaikh, A., and Notare, M. (2006). Towards Building a Highly-Available Cluster Based Model for High Performance Computing, International Parallel and Dis- trib- uted Processing Symposium. 1-8

work page 2006
[12]

V., Chakka., R., and Ever, E

Gemikonakli, O., Do, T. V., Chakka., R., and Ever, E. (2005). Numerical Solution to the Performability of a Multiprocessor System with Reconﬁguration and Rebooting Delays. In Proceedings of ECMS. 766-773

work page 2005
[13]

and Gemikonakli, E

Ever, E., Gemikonakli, O., Kocyigit, A. and Gemikonakli, E. (2013). A hybrid approach to minimize state space explosion problem for the solution of two stage tandem queues, J. Network and Computer Applications. 36. 2. 908-926

work page 2013
[14]

Kirsal, Y., Ever, E., Kocyigit, A., Gemikonakli, O., Mapp, G. (2015). Modeling and anal- ysis of vertical handover in highly mobile environments: J. Supercomput. 71. 4352- 4380

work page 2015
[15]

Gemikonakli, O., Chakka, R

Ever, E. Gemikonakli, O., Chakka, R. (2009). Analytical modelling and simulation of small scale. typical and highly available Beowulf clusters with breakdowns and repairs, Simulation Modelling Practice and Theory. 17. 327-347

work page 2009
[16]

Academic (UB-HPC) Compute Cluster Hardware Specs, The University of Buﬀalo, https://www.buffalo.edu/ccr/support/researchfacilities/generalcompute/cluster? hardware?specs.html

work page
[17]

High Performance Computing using Beowulf clusters http://www2.hawaii.edu/zinner/ 101/students/MitchelBeowulf/cluster.html

work page
[18]

Leangsuksun, C., Scott, S.L

Haddad, I. Leangsuksun, C., Scott, S.L. (2003). HA-OSCAR: the birth of highly available OSCAR, Linux Journal, 1-115

work page 2003
[19]

Thanakornworakij, T. (2012). High availability on cloud with HA-OSCAR, Parallel Pro- cessing Workshops. Springer Berlin Heidelberg

work page 2012
[20]

Buyya, R. (1999). High Performance Cluster Computing: Architectures and Systems. Pren- tice Hall PTR, Upper Saddle River, NJ, USA

work page 1999
[21]

and Vojislav, B

Khazaei, H., Misic, J. and Vojislav, B. M. (2012) Performance Analysis of Cloud Com- puting Centers Using M/G/m/m + r Queuing Systems. IEEE Transactions on Parallel And Distributed Systems. 23(5). 936-943

work page 2012
[22]

User Guide

Amazon Elastic Compute Cloud (2010). User Guide. API Version ed. Amazon Web Service LLC or Its Aﬃliate. http://aws.amazon. com/documentation/ec2

work page 2010
[23]

Papadopoulos, Philip M. (2011). Extending clusters to Amazon EC2 using the Rocks toolkit. International Journal of High Performance Computing Applications. 25(3). 317- 327

work page 2011
[24]

[http://jobsearchtech.about.com/od/ historyoftechin- dustry/a/cloud computing.htm] [Updated: 30 August 2015]

Tech: What is Cloud Computing. [http://jobsearchtech.about.com/od/ historyoftechin- dustry/a/cloud computing.htm] [Updated: 30 August 2015]

work page 2015
[25]

Grimmett, G., and Stirzaker, D. (2010). Probability and Random Processes. Third ed. Oxford Univ. Press. 28 Yonal Kirsal, Yoney Kirsal Ever

work page 2010
[26]

Jin, Y., Wen, Y., and Zhang, W. (2014). Content Routing and Lookup Schemes Using Global Bloom Filter for Content-Delivery-as-a-Service. IEEE Systems Journal. 8(1). 268- 278

work page 2014
[27]

Vilaplana, J., Solsona, F., Teixid, I., Mateo, J., Abella, F., and Rius, J. (2014). A queuing theory model for cloud computing. The Journal of Supercomputing. 1-16

work page 2014
[28]

Xiong, K., Perros, H. (2009). Service performance and analysis in cloud computing, In Proceedings of IEEE World Conference Services. 693-700

work page 2009
[29]

(2009) Performance Evaluation of cloud service considering fault recovery

Yang, B., Tan, F., Dai, Y., and Guo, S. (2009) Performance Evaluation of cloud service considering fault recovery. In Proceedings of the First International conference on cloud computing. 571-576

work page 2009
[30]

Vilaplana, J. (2013). The cloud paradigm applied to e-Health. BMC medical informatics and decision making, 13(1)

work page 2013
[31]

and Nassar, R

Song, H., Leangsuksun, C. and Nassar, R. (2006). Availability Modeling and Evaluation on High-Performance Cluster Computing Systems. Journal of Research and Practice in In- formation Technology. 38. 4. 317-335

work page 2006
[32]

(2006) Approximate Solution for 2-Dimensional Markov Processes Modelling Multi-server Systems Prone To Breakdowns, MSc

Sanei, H. (2006) Approximate Solution for 2-Dimensional Markov Processes Modelling Multi-server Systems Prone To Breakdowns, MSc. Thesis, Middlesex University, UK

work page 2006
[33]

Banks, J., Carson, J., and Nelson, B. (2000). Discrete-Event System Simulation. Prentice Hall Englewood Cliﬀs. NJ. USA

work page 2000

[1] [1]

Hwang, K., Geoﬀrey, C., Fox, J. J. Dongarra (2012). Distributed and Cloud Computing: From Parallel Processing to the Internet of Things. Morgan Kaufmann

work page 2012

[2] [2]

Ngxande, M., Moorosi, N. (2014). Development of Beowulf cluster to perform large datasets simulations in educational institutions. Int. J. Computer App., 99, 29-35. Title Suppressed Due to Excessive Length 27

work page 2014

[3] [3]

Ever, E., Gemikonakli, O., Chakka, R. (2006). A Mathematical Model for Highly Avail- able Clusters with One Head and Several Identical Computing Nodes. In Proceedings of the 9th International Conference on Computer Modelling and Simulation. 32-37

work page 2006

[4] [4]

and Plourde, J

Pijanowski, B.C., Tayyebi, A., Doucette, J., Pekin, B.K., Braun, D. and Plourde, J. (2014). A big data urban growth simulation at a national scale: conﬁguring the GIS and neural network based land transformation model to run in a high performance computing (HPC) environment. Environmental Modelling and Software. 51. 250-268

work page 2014

[5] [5]

Song, A., Wang, W., and Luo, J. (2014). Stochastic modeling of dynamic power manage- ment policies in server farms with setup times and server failures. International Journal of Communication Systems. 27. 680-703

work page 2014

[6] [6]

Ekanayake, J., and Geoﬀrey, F. (2010). High performance parallel computing with clouds and cloud technologies. Cloud Computing. Springer Berlin Heidelberg. 20-38

work page 2010

[7] [7]

Ever, E., Gemikonakli, O., Chakka, R. (2006). A Mathematical Model for Performability of Beowulf Clusters. In Proceedings of 39th Annual Symposium on Network Simulation. 118-126

work page 2006

[8] [8]

B., Sterling, T., Savarese, D., Dorband, J., E., Ranawake, U., A., Packer, and Charles., P

Donald, J. B., Sterling, T., Savarese, D., Dorband, J., E., Ranawake, U., A., Packer, and Charles., P. V. (1995). Beowulf: A parallel workstation for scientiﬁc computation. In Pro- ceedings International Conference on Parallel Processing

work page 1995

[9] [9]

and Vos, D

Adams, J. and Vos, D. (2002). Small-College Supercomputing: Building A Beowulf Cluster At A Comprehensive College. In Proceedings of 33rd Technical symposium on computer science education. Cincinnati. Kentucky. 411-415

work page 2002

[10] [10]

Qin, C., Z. (2014). A strategy for raster-based geocomputation under diﬀerent parallel computing platforms. International Journal of Geographical Information Science. 28. 11. 2127-2144

work page 2014

[11] [11]

Boukerche, A., Shaikh, A., and Notare, M. (2006). Towards Building a Highly-Available Cluster Based Model for High Performance Computing, International Parallel and Dis- trib- uted Processing Symposium. 1-8

work page 2006

[12] [12]

V., Chakka., R., and Ever, E

Gemikonakli, O., Do, T. V., Chakka., R., and Ever, E. (2005). Numerical Solution to the Performability of a Multiprocessor System with Reconﬁguration and Rebooting Delays. In Proceedings of ECMS. 766-773

work page 2005

[13] [13]

and Gemikonakli, E

Ever, E., Gemikonakli, O., Kocyigit, A. and Gemikonakli, E. (2013). A hybrid approach to minimize state space explosion problem for the solution of two stage tandem queues, J. Network and Computer Applications. 36. 2. 908-926

work page 2013

[14] [14]

Kirsal, Y., Ever, E., Kocyigit, A., Gemikonakli, O., Mapp, G. (2015). Modeling and anal- ysis of vertical handover in highly mobile environments: J. Supercomput. 71. 4352- 4380

work page 2015

[15] [15]

Gemikonakli, O., Chakka, R

Ever, E. Gemikonakli, O., Chakka, R. (2009). Analytical modelling and simulation of small scale. typical and highly available Beowulf clusters with breakdowns and repairs, Simulation Modelling Practice and Theory. 17. 327-347

work page 2009

[16] [16]

Academic (UB-HPC) Compute Cluster Hardware Specs, The University of Buﬀalo, https://www.buffalo.edu/ccr/support/researchfacilities/generalcompute/cluster? hardware?specs.html

work page

[17] [17]

High Performance Computing using Beowulf clusters http://www2.hawaii.edu/zinner/ 101/students/MitchelBeowulf/cluster.html

work page

[18] [18]

Leangsuksun, C., Scott, S.L

Haddad, I. Leangsuksun, C., Scott, S.L. (2003). HA-OSCAR: the birth of highly available OSCAR, Linux Journal, 1-115

work page 2003

[19] [19]

Thanakornworakij, T. (2012). High availability on cloud with HA-OSCAR, Parallel Pro- cessing Workshops. Springer Berlin Heidelberg

work page 2012

[20] [20]

Buyya, R. (1999). High Performance Cluster Computing: Architectures and Systems. Pren- tice Hall PTR, Upper Saddle River, NJ, USA

work page 1999

[21] [21]

and Vojislav, B

Khazaei, H., Misic, J. and Vojislav, B. M. (2012) Performance Analysis of Cloud Com- puting Centers Using M/G/m/m + r Queuing Systems. IEEE Transactions on Parallel And Distributed Systems. 23(5). 936-943

work page 2012

[22] [22]

User Guide

Amazon Elastic Compute Cloud (2010). User Guide. API Version ed. Amazon Web Service LLC or Its Aﬃliate. http://aws.amazon. com/documentation/ec2

work page 2010

[23] [23]

Papadopoulos, Philip M. (2011). Extending clusters to Amazon EC2 using the Rocks toolkit. International Journal of High Performance Computing Applications. 25(3). 317- 327

work page 2011

[24] [24]

[http://jobsearchtech.about.com/od/ historyoftechin- dustry/a/cloud computing.htm] [Updated: 30 August 2015]

Tech: What is Cloud Computing. [http://jobsearchtech.about.com/od/ historyoftechin- dustry/a/cloud computing.htm] [Updated: 30 August 2015]

work page 2015

[25] [25]

Grimmett, G., and Stirzaker, D. (2010). Probability and Random Processes. Third ed. Oxford Univ. Press. 28 Yonal Kirsal, Yoney Kirsal Ever

work page 2010

[26] [26]

Jin, Y., Wen, Y., and Zhang, W. (2014). Content Routing and Lookup Schemes Using Global Bloom Filter for Content-Delivery-as-a-Service. IEEE Systems Journal. 8(1). 268- 278

work page 2014

[27] [27]

Vilaplana, J., Solsona, F., Teixid, I., Mateo, J., Abella, F., and Rius, J. (2014). A queuing theory model for cloud computing. The Journal of Supercomputing. 1-16

work page 2014

[28] [28]

Xiong, K., Perros, H. (2009). Service performance and analysis in cloud computing, In Proceedings of IEEE World Conference Services. 693-700

work page 2009

[29] [29]

(2009) Performance Evaluation of cloud service considering fault recovery

Yang, B., Tan, F., Dai, Y., and Guo, S. (2009) Performance Evaluation of cloud service considering fault recovery. In Proceedings of the First International conference on cloud computing. 571-576

work page 2009

[30] [30]

Vilaplana, J. (2013). The cloud paradigm applied to e-Health. BMC medical informatics and decision making, 13(1)

work page 2013

[31] [31]

and Nassar, R

Song, H., Leangsuksun, C. and Nassar, R. (2006). Availability Modeling and Evaluation on High-Performance Cluster Computing Systems. Journal of Research and Practice in In- formation Technology. 38. 4. 317-335

work page 2006

[32] [32]

(2006) Approximate Solution for 2-Dimensional Markov Processes Modelling Multi-server Systems Prone To Breakdowns, MSc

Sanei, H. (2006) Approximate Solution for 2-Dimensional Markov Processes Modelling Multi-server Systems Prone To Breakdowns, MSc. Thesis, Middlesex University, UK

work page 2006

[33] [33]

Banks, J., Carson, J., and Nelson, B. (2000). Discrete-Event System Simulation. Prentice Hall Englewood Cliﬀs. NJ. USA

work page 2000