Abstract
Simulation studies for Web systems have been carried out by many academic researchers and practitioners. Models are often less time-consuming to develop and run production system. Performance Engineering is done to determine the system performance. In the paper various performance models of Cluster-based Web Systems are discussed, as well as their influence on response time. The Queueing Petri Nets simulations are based on different loads, but also on changing environmental parameters and system structures. A novelty in this approach is the use of two client-classes related to customer behavior and routes in the system. In all cases Web system architectures include clusters are taken into consideration. Simulation results obtained from this models are compared with data from a real system and show good accuracy.
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Notes
- 1.
We have the same number corresponding to each other logical layers and physical tiers.
- 2.
We assumed that the reader is familiar with PE and QPN formalism.
- 3.
We analysed queueing systems with the Poisson clients arrival process.
- 4.
We analysed queueing systems with the exponential clients service process.
- 5.
We used IS for clients station, PS for FE servers and FIFO for BE server.
- 6.
This paper considers single server queue.
- 7.
Size of the queue is infinite.
- 8.
PE analyzes the expected performance characteristics of system during the different phases of its life cycle.
- 9.
Classes are distinguished by different routing probabilities of requests in the model.
- 10.
Departure disciplines are an extension to the QPN modeling formalism: NORMAL implies that tokens become available for output transitions immediately upon arrival, FIFO implies that tokens become available for output transitions in the order of their arrival, i.e., a token can leave the place/depository only after all tokens that have arrived before it have left. The departure discipline of an ordinary place or depository determines the order in which arriving tokens become available for output transitions.
- 11.
Inter-arrival time (arrival process) \(1/\lambda _i\) with mean arrival rate per unit time parameter \(\lambda _i\).
- 12.
Service time \(1/\mu _i\) with service rate per unit time parameter \(\mu _i\).
- 13.
A color specifies a type of tokens that can be resided in the place.
- 14.
The subscript MS denotes multisets. \(CO(p)_{MS}\) denotes the set of all finite multisets of CO(p).
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A Appendix: Mathematical Queueing Petri Nets Model
A Appendix: Mathematical Queueing Petri Nets Model
QPN is an tuple (16), where CPN is Colored Petri Net (15) [7].
where:
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\(PL=\{p_1,p_2,...,p_i\}\) is a finite and non-empty set of places,
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\(TR=\{t_1,t_2,...,t_j\}\) is a finite and non-empty set of transitions,
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\(PL \cap TR = \varnothing \),
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CO is a color function defined from \(PL \cup TR\) into finite and non-empty sets (specify the types of tokens that can reside in the place and allow transitions to fire in different modes),
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IN(p, t) are the backward and forward incidence functions defined on \(PL \times TR\), such that \(IN(p,t) \in [CO(t) \longrightarrow CO(p)_{MS}], \forall (p,t) \in PL \times TR\)Footnote 14 (specify the interconnections between places and transitions),
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MA(p) is an initial marking defined on PL such that \(MA(p) \in CO(p), \forall p \in PL\) (specify how many tokens are contained in each place).
where:
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\(QU=(QU_1,QU_2,(q_1,...,q_{|PL|}))\), where:
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\(QU_1 \subseteq PL\) is a set of timed queueing places,
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\(QU_2 \subseteq PL\) is a set of immediate queueing places,
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\(QU_1 \cap QU_2 = \varnothing \),
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\((q_1,...,q_{|PL|})\) is an array with description of places (if \(p_i\) is a queueing place \(q_i\) denotes the description of a queue with all colors of \(CO(p_i)\) into consideration or if \(p_i\) is the ordinary place (\(p_i\)) equals null).
-
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\(WE=(WE_1,WE_2,(w_1,...,w_{|TR|}))\), where:
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\(WE_1 \subseteq TR\) is a set of timed transitions,
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\(WE_2 \subseteq TR\) is a set of immediate transitions,
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\(WE_1 \cap WE_2 = \varnothing \), \(WE_1 \cup WE_2 = TR\),
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\((w_1,...,w_{|TR|})\) is an array (entry \(w_j \in [CO(t_j) \longmapsto \mathbb {R}^{+}]\) such that \(\forall c \in CO(t_j) : w_j(c) \in \mathbb {R}^{+}\)) of:
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\(*\) rate of a negative exponential distribution specifying the firing delay due to color, if \(t_j \in WE_1\),
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\(*\) firing weight specifying the relative firing frequency due to color, if \(t_j \in WE_2\).
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Based on definition (16) we define following QPN model (17) of CWS.
where:
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\(PL=\{FE, BE, ThreadsPool, ConnectionsPool\}\),
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\(TR=\{t_1,t_2,t_3, t_4,t_5\}\),
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\(CO(p_i)\) for \(c=\{K_1, K_2, tp, cp\}\), where:
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\(K_1=250\) and \(K_2=250\) - client-classes,
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\(th=\{30, 60, 90\}\) - threads,
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\(cp=40\) - connections,
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IN(p, t),
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\(MA(p)=\{Clients(250,250), ThreadsPool(30, 60, 90), ConnectionsPool(40)\}\),
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\(QU=(QU_1, QU_2, (-/M/\infty /IS_{Clients}, null,\)
\(-/M/1/PS_{Sub-FE}, null,\)
\(-/M/1/FIFO_{Sub-BE}, null, null))\), where:
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\(QU_1=\{Clients, FE\_CPU_n, BE\_I/O\}\), where \(n=\{1, 2, 3\}\),
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\(QU_2= \varnothing \),
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\(WE=(WE_1,WE_2)\), where:
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\(WE_1=\varnothing \),
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\(WE_2=TR\),
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\(\forall c \in CO(t_j) : w_j(c):=1\) (all transition firings are equally likely).
-
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Rak, T. (2019). Cluster-Based Web System Models for Different Classes of Clients in QPN. In: Gaj, P., Sawicki, M., Kwiecień, A. (eds) Computer Networks. CN 2019. Communications in Computer and Information Science, vol 1039. Springer, Cham. https://doi.org/10.1007/978-3-030-21952-9_26
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