Simulation of Reliability in Multi-server Computer Networks

Transcription

1 Simulation of Reliability in Multi-server Computer Networks SAULIUS MINKEVIČIUS, VU Institute of Mathematics Informatics, Akademijos 4, Vilnius, Lithuania Vilnius University Mathematics Informatics faculty, Naugarduko 24, LT Vilnius, LITHUANIA Abstract: -The performance in terms of reliability of multi-server computer networks motivates this paper. The probability limit theorem is derived on the virtual waiting time of job in nodes in a multi-server open queueing network applied to a reliability model for multi-server computer networks we relate the time of failure of a multi-server computer network to the system parameters. Key-Words: - Performance evaluation, reliability theory, queuing theory, mathematical models of technical systems, multi-server open queuing network, heavy traffic, the probability limit theorem, a virtual waiting time of job in nodes in a multi-server open queuing network. 1 Introduction Probabilistic models queuing networks have long been used to study the performance reliability of computer systems [4, 5] to analyse the performance reliability of computer networks of distributed information systems [6, 7]. In this paper, we will first review the works related to using the queuing theory of computer systems reliability, then present some new results on the estimation of the time of failure of a computer network. In one of the first papers of this kind [20], an an efficient algorithm was investigated for the reliability of a circular consecutive-k-out-of-n system. The author in paper [10] considers network topological optimization with a reliability constraint. In [9] the classical twoterminal reliability is extended for a wireless computer network. Papers [13, 1] investigated the reliability of distributed systems computer networks in which computing nodes communication links may fail with certain probabilities. In [12] limit theorems on the extreme queue length extreme virtual waiting time in open queuing networks in heavy traffic are derived applied to a reliability model for computer networks. In [18, 19], the primitives, concepts, terminology, fundamental ideas of service reliability theory engineering were developed. Also, the purpose of these papers is to describe some mathematical models useful for service reliability engineering to illustrate the use of these ideas by some examples. Paper [3] investigated heavy traffic asymptotics for systems with unreliable servers. In this paper, we present the probability limit theorem on the virtual waiting time of job in nodes of a multi-server open queuing network in heavy traffic conditions. 2 The Network Model Consider a network of stations, indexed by The station has servers, indexed by. A description of the primitive data construction of processes of interest are the focus of this section. No probability space will be mentioned in this section, of course, one can always think that all the variables processes are defined on the same probability space. First, are sequences of exogenous interarrival times, is the interarrival time between the job the st job which arrive at the station j exogenously (from the outside of the network). Define is called the exogenous arrival process of the station j, i.e., counts the number of jobs that arrived at the station from the outside of the network. ISBN:

2 Second, are service times, sequences of is the service time for the th customer served by the server of the station Define is called a service process for the server at the station, i.e., counts the number of services completed by the server at the station during the server s busy time. We note with all of these terms assumed finite. In addition, let be the total number of jobs routed to the th station of the network in the interval be the total number of jobs after service departure from the th station of the network in the interval be the total number of jobs routed to the server of the th station of the network in the interval let be the total number of customers after service departure from the server of the th station of the network in the interval be the total number of jobs after service departure from the server of the th station of the network routed to the server of the th station of the network in the interval Let be a probability of the job after service at the th station of the network routed to the th station of the network. Denote as part of the total number of jobs which, after service at the server of the th station of the network, are routed to the server of the th station of the network in the interval Also let as the total amount of time that a server at the station is busy in the interval Now we introduce the following processes, indicates the number of jobs (queue of length) waiting to be served by the server of the station at time Now we introduce the following processes, indicates the virtual waiting time of job (workload process) in server of the station at time ; The dynamics of the queuing system (to be specified) depends on the service discipline at each service station. To be more precise, first come, first served (FCFS) service discipline is assumed for all stations. When a customer arrives at a station finds more than one server available, it will join one of the servers with the smallest index. We assume that the service station is work conserving; namely, not all servers at a station can be idle when there are customers waiting for service at that station. In particular, we assume that a station must serve at its full capacity when the number of jobs waiting is equal to or exceeds the number of servers at that station. 3 The Main Results Now we present probability limit theorem for the virtual waiting time of job in nodes of a multi-server open queuing network. Similarly as in [11], let us investigate multi-server open queuing networks in a series: are independent identically distributed rom variables in the th network;. Also, we define Let us denote We will also assume that the following overload conditions" are fulfilled. (1) ISBN:

3 Note that these conditions quarantee that the length of all the queues will grow indefinitely with probability one. The results of the present paper are based on the following theorem: Theorem 3.1. If conditions (1) are fulfilled, then Proof. Applying the results of paper [2], we see that virtual waiting time of job (workload process) in server of station is equal Denote (2) Proof. At first, using Theorem 3.1, we get that for Let us investigate a computer network node, which consists of elements (computers), Denote Note that Denote the structural function of the system of elements connected by scheme 1 from (see, for example, [13], as follows: At first we will prove that At last we have that (3) of the computer node. Suppose is the parameter of failure (4) are stard Wiener processes, Thus, using the convergence together theorem from (3) (4) we derive that is stard Wiener processes, By [11], for the vector convergence to hold it suffices that a limit process is continuous in Using this note we complete the proof of the theorem. Let us estimate the reliability function of the system (computer network) using the formula of a conditional probability for the reliability function of the computer node Note that But for fixed we have that 4 The Reliability of a Multi-server Computer Network Theorem 4.1. If conditions (1) are fulfilled, the computer network becomes unreliable (all computers fail). ISBN:

4 Finally, we denote as a probability of blocking (workload process) of the server in node in a multi-server computer network, So, is the probability of blocking of the whole multi-server computer network (because if Thus, (5) Applying Theorem 3.1, we obtain that for then for all ),. Thus, we prove the following corollary about the probability of blocking of a multi-server computer network. Corollary 5.1. If conditions (1) are fulfilled, then (6) 6 Proof. Denote we achieve that as a complement of set A. Thus, It follows from (6) that (7) Consequently, (see (4) (6)), which completes the proof. (8) 5 Applications of the Main Results At first we present a theorem about the fluid approximation for the virtual waiting time of job in nodes of a multi-server open queuing network under heavy traffic conditions. Theorem 5.1. If conditions (1) are fulfilled, then If conditions (1) are satisfied, let us prove that. Next, we present the law of the iterated logarithm for the virtual waiting time of job in nodes of a multi-server open queuing network under heavy traffic conditions. Theorem 5.2. If conditions (1) are satisfied, then But for applying Theorem 3.1 we obtain (9) is the normal distribution function. Consequently, we derive that (see (7) (9)) ISBN:

5 or The proof of Corollary 5.1 is complete. Consequently, if conditions (1) are fulfilled, then the whole multi-server network is busy. Consequently, if conditions (1) are fulfilled, then the whole multi-server network is busy. 6 Concluding remarks future research 1. If the conditions of the theorem on the probability limit theorem are fulfilled (i. e., conditions (1) are satisfied), the network is occupied at first (see Corollary 4.1) if conditions (1) are satisfied later on, the network becomes uncontrollable after a certain time (as ) (see Corollary 5.1). 2. Conditions (1) are fundamental, - the behaviour of the whole network its evolution is not clear, if conditions (1) are not satisfied. Therefore, this fact is the object of further research discussion. 3. Note that the computer with the Windows opera- =tion system functions steadily, if the number of jobs does not exceed 5 (therefore,. In the opposite cases, the computer fails (see paragraph 1). 4. The theorems of this paper are proved for a class of a multi-server open queuing network in heavy traffic with the service principle ``first come, first served", endless waiting time of a customer in each node of the queuing system, the times between the arrival of customers at the multi-server open queuing networks are independent identically distributed rom variables. However, analogous theorems can be applied to a wider class of multiserver open queuing networks in heavy traffic: when the arrival service of customers in a queue is by groups, when interarrival times of customers at the multi-server open queuing network are weakly dependent rom variables, etc. References: [1] Yinong Chen, Zhongshi He,.Bounds on the reliability of distributed systems with unreliable nodes links, IEEE Transactions on Reliability, Vol.53, No.2, 2004, pp [2] H. Chen H., Shanthikumar J. George. Fluid limits diffusion approximations for networks of multi-server queues in heavy traffic, Discrete Event Dynamic Systems, Vol.4, 1994, pp [3] Jean Francois Dantzer, Isi Mitrani, Philippe Robert. Large scale heavy traffic asymptotics for systems with unreliable servers, Institut National De Recherche En Informatique Et En Automatique}, 3807, November [4] E. Gelenbe, Unified approach to the evaluation of a class of replacement algorithms, IEEE Transactions on Computers, Vol. C22, No. 6, pp [5] E. Gelenbe, Probabilistic models of computer systems, Part II, Acta Informatica, Vol,12, pp , [6] E. Gelenbe, D. Finkel, S.K. Tripathi Availability of a distributed computer system with failures, Acta Informatica, 23 (6): , November [7] E. Gelenbe, X.W. Wang., R. Onvural Diffusion based statistical call admission control in ATM, Performance Evaluation}, Vol.27, No.8, 1996, pp [8] E. Gelenbe, J.M. Fourneau, G-networks with resets, Performance Evaluation, Vol.49, No.(1-4), 2002, pp [9] Sahin Yasar. Algorithm design for reliability analysis in mobile communication networks, Journal Of Aeronautics And Space Technologies, Vol. 3, No.1, 2007, pp [10] Rong Hong Jan, Fung Jen Hwang, Sheng Tzong Cheng, Topological optimization of a communication network subject to a reliability constraint, IEEE Transactions on Reliability, Vol.42, No. 1, 1993, pp [11] S. Minkevičius, Weak convergence in multiphase queues, Lietuvos Matematikos Rinkinys, Vol. 26, 1986, pp , (in Russian). [12] S. Minkevičius, Analysis of the componentbased reliability in computer networks, CUBO A Mathematical Journal, Vol. 12, No. 1, 2010, pp [13] H. Mohamed, S. Mohamed, Yang Xiaozong, Liu Hongwei Wu Zhibo, An efficient algorithm for reliability upper bound of distributed systems with unreliable nodes, International Journal on Computer Science Engineering, Vol. 02, No 02, 2010, pp [14] J.J. Morder, S. E. Elmaghraby (eds.), Hbook of operational research models applications, Van Nostr Reinhold, New York, [15] L.L. Sakalauskas, S. Minkevičius. On the law of the iterated logarithm in open queueing ISBN:

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