Cluster-Based Web System Models for Different Classes of Clients in QPN

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.

A Appendix: Mathematical Queueing Petri Nets Model

QPN is an tuple (16), where CPN is Colored Petri Net (15) [7].

$$\begin{aligned} CPN=(PL, TR, CO, IN, MA) \end{aligned}$$

(15)

where:

• $PL=\{p_1,p_2,...,p_i\}$ is a finite and non-empty set of places,

• $TR=\{t_1,t_2,...,t_j\}$ is a finite and non-empty set of transitions,

• $PL \cap TR = \varnothing$,

• 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),

• IN(p, t) are the backward and forward incidence functions defined on $PL \times TR$, such that $IN(p,t) \in [CO(t) \longrightarrow CO(p)_{MS}], \forall (p,t) \in PL \times TR$¹⁴ (specify the interconnections between places and transitions),

• 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).

$$\begin{aligned} QPN=(CPN, QU, WE) \end{aligned}$$

(16)

where:

• $QU=(QU_1,QU_2,(q_1,...,q_{|PL|}))$, where:

  • $QU_1 \subseteq PL$ is a set of timed queueing places,

  • $QU_2 \subseteq PL$ is a set of immediate queueing places,

  • $QU_1 \cap QU_2 = \varnothing$,

  • $(q_1,...,q_{|PL|})$ 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).

• $WE=(WE_1,WE_2,(w_1,...,w_{|TR|}))$, where:

  • $WE_1 \subseteq TR$ is a set of timed transitions,

  • $WE_2 \subseteq TR$ is a set of immediate transitions,

  • $WE_1 \cap WE_2 = \varnothing$, $WE_1 \cup WE_2 = TR$,

  • $(w_1,...,w_{|TR|})$ 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:

    • $*$ rate of a negative exponential distribution specifying the firing delay due to color, if $t_j \in WE_1$,

    • $*$ firing weight specifying the relative firing frequency due to color, if $t_j \in WE_2$.

Based on definition (16) we define following QPN model (17) of CWS.

$$\begin{aligned} QPN=(PL, TR, CO, IN, MA, QU, WE) \end{aligned}$$

(17)

where:

• $PL=\{FE, BE, ThreadsPool, ConnectionsPool\}$,

• $TR=\{t_1,t_2,t_3, t_4,t_5\}$,

• $CO(p_i)$ for $c=\{K_1, K_2, tp, cp\}$, where:

  • $K_1=250$ and $K_2=250$ - client-classes,

  • $th=\{30, 60, 90\}$ - threads,

  • $cp=40$ - connections,

• IN(p, t),

• $MA(p)=\{Clients(250,250), ThreadsPool(30, 60, 90), ConnectionsPool(40)\}$,

• $QU=(QU_1, QU_2, (-/M/\infty /IS_{Clients}, null,$

  $-/M/1/PS_{Sub-FE}, null,$

  $-/M/1/FIFO_{Sub-BE}, null, null))$, where:

  • $QU_1=\{Clients, FE\_CPU_n, BE\_I/O\}$, where $n=\{1, 2, 3\}$,

  • $QU_2= \varnothing$,

• $WE=(WE_1,WE_2)$, where:

  • $WE_1=\varnothing$,

  • $WE_2=TR$,

  • $\forall c \in CO(t_j) : w_j(c):=1$ (all transition firings are equally likely).