modelling open networks with breakdowns, repairs and finite...

Modelling Open Networks with Breakdowns, Repairs and Finite Buffers, Using an IPP Departure Process Model

RAM CHAKKA1, ENVER EVER2, and ORHAN GEMIKONAKLI2

1RGMCET, Nandyal, India 518501 2School of Computing Science, Middlesex University, London NW4 4BT, UK

Abstract: Open queuing network systems are useful in the computer industry. In this paper, a multi-node open network, with heterogeneous nodes, each node serving external as well as routed internal arrivals of jobs is considered. The nodes are prone to failures and repairs, the buffers are of finite size. An approximate performability model is developed based on an IPP departure process. External Poisson arrivals and routed internal arrivals are modelled together using an MMPP model. Results have been validated via simulation. Keywords: Queuing, Markov Modulated Poisson Process (MMPP), Interrupted Poisson Process (IPP), Open Queuing Networks, Breakdowns and Repairs. 1 Introduction Open network systems with breakdowns and repairs are very useful to model various kinds of practical computer and communication networks [2, 11]. In this paper we develop an approach to model open network systems with breakdowns, repairs, and with finite buffers, by suitably extending a previously reported performance model with unbounded buffers [3]. At each node, it is possible to approximate the traffic as the superposition of external arrivals and internal arrivals fed by other nodes, by considering it as a single stream with bursty nature. In [7, 12, 14] various approaches are given to model systems with bursty arrivals. However, in these approaches, server breakdowns and repairs are not considered. In [13] the batch Markov arrival process (BMAP) is used as a mathematical model for describing bursty traffic, in a single server system with the server subject to breakdowns and repairs. In [3] an approximate performance model based on the use of Markov Modulated Poisson Process (MMPP) to represent various bursty arrival and departure processes is presented. In this paper, we derive an approximate model and solution for the steady state probabilities of open network systems with breakdowns and repairs, and with bounded queuing capacities, by developing an efficient iterative procedure that employs the spectral expansion solution algorithm as a major computation tool, in each iteration.

Spectral expansion is an exact solution method for the steady state analysis of certain two-dimensional Markov processes, known as QBD and QBD-M processes [1, 6]. These Markov models have been used widely and successfully, in performance and availability analysis of computing and communication systems. Performability modelling of multiprocessor clusters with bounded and unbounded job queues [1, 8, 10], open networks of queues with unreliable servers [6] and performance evaluation of many other practical systems were considered.

The Markov models of many practical systems have finite buffer space, rather than an infinite one. In this paper we use several other engineering techniques together with the spectral expansion method in order to find an approximate solution for the performability analysis of open networks with bounded buffers, breakdowns and repairs. Numerical performability results have been presented for such systems. Simulation results have also been obtained to confirm the degree of accuracy for the models considered.

The paper is organised as follows: The next section describes the open network system with finite buffers, breakdowns and repairs, considered in this work. Exact solution for steady state performability using the spectral expansion method is briefly covered in the section on steady state solution. Departure process model used in this study is explained in turn. Approximate solution to these systems and numerical results are then presented.

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp33-38)

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2 The System Under Study

The system analyzed is an open network with K nodes where each node contains a server and a bounded FIFO queue. In such a system the total flow of jobs is the superposition of external (Poisson) and internal flows from other nodes. In many practical systems, the arrival rates may vary randomly over time, quite sharply sometimes. Examples include superposition of packet oriented voice processes, and packet data in communication modelling [2]. In this paper, to represent the traffic into a node consisting of Poisson external arrivals together with feedbacks expected from other nodes, an MMPP is used.

Each node in the system is then treated as an MMPP/M/1/L queuing model with breakdowns and repairs. When a server breaks down, it is set into repair process immediately. After the completion of the repair, the server becomes operative again. We use the Interrupted Poisson process for departure process modelling of these nodes. This approach has been used for performability evaluation of open queuing systems with unbounded queuing capacities in [3, 6], with some success. However, open queuing systems with breakdowns, repairs and bounded queuing capacities are not considered so far.

Figure 1: An open queuing network with 3=K 3 The Steady State Solution The steady state solution for the proposed system is obtained following the spectral expansion solution [1]. Once the steady state probabilities are computed, a number of steady-state performability measures such as availability, reliability, loss rate of jobs, can be computed using well-known formulas. For illustration, we have concentrated on the mean queue length (MQL) and the percentage of jobs lost (PJL), only. The service, breakdown and repair times of the

servers are exponentially distributed with rates µk, ξk, and ηk, respectively. The queuing capacity of each node is limited. Each server serves one job at a time when it is operative. The external arrival rate to node k is σk. When a job’s service is interrupted because of a server breakdown, its service is resumed when a server is available, on the basis of resume or repeat with re-sampling discipline. After a job is serviced at node k, it is routed to node l with a probability of qk,l . It is assumed that qk,k =0 (k=1, 2, …, K). The probability of a job to leave the system after being

served at node k is q . A square

matrix Q of size KxK is formed as, Q(k,l) = qk,l (k,l = 1, 2 ,…, K). Q is the routing probability matrix strictly among the nodes. This system is shown in Figure 1, for K=3. Various steady state performance measures can be considered.

∑=

+ −=K

1lk,1Kk, q1

l

4 Departure Process Modelling In networks of queues the departed jobs are fed to other relevant queues, as arrivals for further servicing. Hence, departure process modelling is an important issue for analyzing open networks. In this paper the Interrupted Poisson process has been chosen as the model for the departure process. IPP can be defined as a two-phase MMPP where flow rate in one of these phases is equal to 0 [9]. Let the parameters of this departure process be ν,α, and β. In phase 1 the departure rate is υ. In phase 2 the departure rate is 0. Transition rate from phase 1 to phase 2 is α, and from phase 2 to phase 1 is β. Computation of these parameters is done by computing the moments of the inter-departure times, from the parameters of the system, and working out the parameters from these computed moments. For this purpose the algorithm given in [2, 3, 6] has been followed and IPP departure model 1 has been chosen for departure process modelling of open networks with breakdowns, repairs and finite queuing capacities. In this model, ν, α, and β are defined as follows:

Clearly, the system described above is important for communication and computer networks [2, 11]. We develop an iterative solution that uses the spectral expansion method, to solve such queuing problems for approximate steady state probabilities.


;)32)(66(

)2(92231321

31

3212

ddddddddd

−+−−

=α ; 66

)2(3

32131

212

dddddd

+−−

=β kkkgkkk Ik

σ+Σ⊕⊕Σ⊕Σ=Σ ),(),2(),1( K

),(),2(),1( , kgkkk kΘ⊕⊕Θ⊕Θ=Θ L

where, ⊕ stands for Kronecker sum, σk is the arrival rate of external arrivals to node k, and Ik is the unit matrix with the same size as Σk.

2231

32131

32)66(2

ddddddd

−+−

=ν ,

where d1,d2, and d3 are the first three moments of the random variable, representing the inter-arrival times of the departure process. A procedure is given in [2, 6] to compute d1, d2 and, d3 accurately.

With the above procedures to compute MMPPks and IPPk,ls, it is possible to use an iterative solution [3] to obtain the mean queue length measures of each node in the system. This procedure can be summarised as follows:

1. In the first step the total average arrival rates to all

nodes are computed. Each node is solved as an MMPP/M/1/L queue with breakdowns and repairs by using the computed average arrival rates and the spectral expansion method for bounded queuing systems. In this step, MMPPks are considered as Poisson processes (with average arrival rate in each phase). Also in this step queue lengths Lks and departure process parameters are computed.

5 The Solution for the Proposed Model An iterative procedure similar to the one given in [2, 3, 6] can be used to solve the model representing open queuing networks with breakdowns, repairs and finite queuing capacity, for different performance evaluation measures. For this study we have selected independent Poisson processes for external arrivals to the nodes. The departures from any node are approximated to an IPP following the IPP model 1 in [3, 6]. To be able to use an IPP model for a departure process, some assumptions have been made [3]. Let Sk be the set of nodes which the departures from node k are routed and Gk be the set of nodes which node k receives jobs from. Then, the split process of the IPP departure from node k, entering node l is also approximated by an IPP with parameters νk,l, αk,l and βk,l. Also, the internal arrival IPPl,ks and the external Poisson arrivals to node k are assumed to be independent for each node k. With these assumptions, node k receives one independent IPP from each node belonging to set Gk plus its independent external Poisson. The superposition of all these independent streams entering node k, is an MMPP with 2 phases, where is the cardinality of Gk. Then, each node can be considered as an MMPP/M/1/L model and solved by following the procedure given in previous sections.

kg

kg

2. In the second step the IPPks computed in previous iteration are used to re-construct the MMPPks. Each node is again solved as a MMPP/M/1/L queue by using the spectral expansion method. New Lk values are calculated. In this step the variable ε is defined as ∑

=

−=k

k

oldkk LL

1

)(ε .

3. If ε, computed in the previous step, is larger than a specified tolerance value then the procedure goes back to step 2.

4. If ε, computed in the previous step, is less than the specified tolerance value then all required steady state performance measures are computed with the most recent parameters.

6 Numerical Results and Discussions In this section numerical results are presented for three server tandem networks with breakdowns and repairs. In all computations the tolerance value is taken as 0.01. Service rates, failure rates and repair rates of nodes 1, 2, and 3 are µ1=5.0, ξ1=0.001, η1=0.002, µ2=6.0, ξ2=0.002, η2=0.003, µ3=7.0, ξ3=0.003, η3=0.005 respectively. In computations the number of iterations needed to obtain a tolerance value less than 0.01 has been found to be between three and five depending on the system parameter values.

Probabilistic splitting of IPP and constructing MMPP are very important issues used in the iterative procedure given. More detailed information about these processes is given in [6, 9].

Let IPPk be the departure process of node k with parameters ν k, α k and β k. Then parameters of IPPk,ls are ν k,l, α k,l and β k,l where ν k,l = q k,lν k , αk,l =α k and βk,l =β k [2, 6].

Let MMPPk be the overall arrival process of node k. Then the matrices Θk and Σk can be expressed as follows:


Finally, in Figure 4 node 1 has been considered to show the percentage of jobs lost for various queuing capacities.

Since the most challenging systems are the ones which have external arrivals to only one of the servers [2], we selected σ2=0, and σ3=0. In the computations performed for Figure 2 the routing probability matrix Q is used where as for Figures 3 and 4, Q(1) is used. Q, and Q(1) matrices are given below. Please note that the K+1th column of these matrices are given to show the probability of a job to leave the system.

=

0.10.00.00.00.00.10.00.00.00.00.10.0

Q

=

5.00.00.05.00.00.10.00.06.00.04.00.0

)1(Q

Figure 2 presents mean queue length values of

node 2 as a function of queuing capacities for various arrival rate values. In Figure 3 mean queue length values have been computed for node 1. Figure 4: Percentage of jobs lost at node 1

Since the solution given is approximate, it is important to assess the accuracy of the models proposed. For this purpose, various simulation results have been obtained for a 3-stage tandem network with various queuing capacities. All other parameters are taken same as previous computations. The routing probability matrix Q is used for both simulation and analytical model computations. Figures 5-10 show these results together with results obtained using the analytical model. It can clearly be seen that the two set of results are closer for node 1 of the tandem network.

Figure 2: MQL values at node 2 as a function of queuing capacities for various arrival rate values

Figure 5: MQL at node 1 and small L values

Figure 3: MQL values of node 1 as a function of σ for Q(1)


Figure 6: MQL for node 1 and large L values Figure 9: MQL for node 3 and small L values

Figure 10: MQL for node 3 and large L values Figure 7: MQL for node 2 and small L values

8 Conclusions and Future Directions In this paper a model and solution technique have been presented for performability analysis of open queuing networks with server breakdowns, repairs, external job arrivals and finite queuing capacity and applied to a three-stage network.

Poisson and IPP have been used for arrival and departure process modelling respectively, at nodes. Each node has been considered as an MMPP/M/1/L queuing model. An iterative procedure that solves MMPP/M/1/L models in each iteration has been developed and numerical results have been obtained. In addition to spectral expansion method, several other engineering techniques have been used in order to compute the probabilistic splitting of the departure process and reconstruction of the arrival processes. Simulation results are also presented for validation and comparison purposes.

Figure 8: MQL for node 2 and large L values


[8] Ever, E., O. Gemikonakli, R.Chakka, and T.V.Do, A Mathematical Model for Performability Evaluation of Heterogeneous Multiprocessor Systems with Reconfiguration and Rebooting delays, Proceedings of ESM 2005, Porto, Portugal, 2005, pp. 487-495.

The proposed model is reasonably accurate for all three stages. We believe that since IPP is a renewal process but the arrival MMPPs are non renewal, it is expected that more accurate results will be achieved by using a 2-phase MMPP model for departure processes.

[9] Fischer W., and K. Meier-Hellstern, The Markov-modulated Poisson Process, Performance Evaluation, Vol.18, No.2, 1993, pp. 149-171.

If the external arrivals are sufficiently bursty, one can use independent MMPPs with 2-phases instead of the Poisson processes, for external arrivals at nodes. Then, the overall arrivals at a node would be represented by an MMPP with 2 number of phases. This of course increases the complexity of the model, but, such a model would take care of the burstiness of external arrivals.

1+kg [10] Gemikonakli O., T.V. Do, R. Chakka, and E. Ever, Numerical Solution to the Performability of a Multiprocessor System with Reconfiguration and Rebooting Delays, Proceedings of ECMS 2005, Riga, Latvia, June 2005, pp. 766-773.

[11] Kelly F.P., Modelling Communication Networks, Present and Future, Proceedings of Philosophical Transactions of the Royal Society, London, U.K., 1996, pp. 437-463.

References: [1] Chakka, R., Spectral Expansion Solution for

Some Finite Capacity Queues, Annals of Operations Research, 79, 1998, pp. 27-44.

[12] Klimenok V.I., A. V. Kazimirsky, A. N. Dudin, L. Breuer, U.R. Krieger, Queueing Model MAP|PH|1|N with Feedback Operating in the Markovian Random Environment, Austrian Journal of Statistics, Vol.34, No.2, 2005, pp.101-110.

[2] Chakka, R., Performance and Reliability Modelling of Computing Systems Using Spectral Expansion, PhD Thesis, University of Newcastle Upon Tyne, UK, 1995 (available and can be obtained from the first author).

[3] Chakka, R., O. Gemikonakli, and P.G. Harrison, Approaches to Modelling Open Networks with Bursty Arrivals, Proceedings of Eighth IFIP Workshop on Performance Modelling and Evaluation of ATM & IP Networks. Ilkley. July, 2000, pp. 13/1-13/11.

[13] Li Q.L., Y. Ying, and Y. Zhao, The BMAP/G/1 Retrial Queue with Server Subject to Breakdowns and Repairs, Annals of Operations Research, Vol.141, 2006, pp. 253-292.

[14] Ny L. M. L, and B. Sericola, Transient Analysis of the BMAP/PH/1 Queue, International Journal of Simulation: Systems, Science & Technology, Special Issue on Analytical & Stochastic Modelling Techniques, Vol.3, No.3-4, 2002, pp. 4-14.

[4] Chakka, R., and I. Mitrani, A Numerical Solution Method for Multiprocessor Systems with General Breakdowns and Repairs, Proceedings of the 6th International Conference Modelling Techniques and Tools, Edinburgh, UK, 1992, pp. 289-304.

[5] Chakka, R., and I. Mitrani, Heterogeneous Multiprocessor Systems with Breakdowns: Performance and Optimal Repair Strategies, Theoretical Computer Science, 125, 1994, pp. 91-109.

[6] Chakka, R., and I. Mitrani, Approximate Solutions for Open Networks with Breakdowns and Repairs, Stochastic Networks: Theory, Oxford University Press, 1996.

[7] Dudin A.N., A.A. Shaban, and V.I. Klimenok, Analysis of a BMAP|G|1|N queue, International Journal of Simulation: Systems, Science and Technology, Vol.6, No.1-2, 2005, pp. 13-23.


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