modelling and simulation of finite markov multi- server ...

MODELLING AND SIMULATION OF FINITE MARKOV MULTISERVER QUEUEING SYSTEM SUBJECT TO BREAKDOWNS DORDA Michal, (CZ), TRESLER Filip, (CZ), TUREK Michal, (CZ), TUREK Richard, (CZ) Abstract. This paper is devoted to modelling and simulation of a finite Markov multi-server queueing system subject to breakdowns and with an ample repair capacity. The paper introduces a mathematical model of the studied system and a simulation model created by using software CPN Tools, which is intended for modelling and a simulation of coloured Petri nets. At the end of the paper the outcomes which were reached by both approaches are presented and statistically evaluated. Key words. M/M/n/m, Queueing system, Breakdown, Petri net Mathematics Subject Classification: Primary 60K25, 90B22; Secondary 68U20.

1

Introduction

The server that is working without failures is usually assumed in the queueing theory. But in many practical cases this assumption is not correct; servers are often technical devices and every technical device can be broken. Many authors studied a behavior of single server queueing systems subject to breakdowns under diverse assumptions. Modeling of multi-server queueing systems is not so often done due to mathematical complexity of their analysis. Mitrani and Avi-Itzak [1] investigated a multi-server Markov queueing model with an infinite queue capacity, server breakdowns and an ample repair capacity. Neuts and Lucantoni [2] and Wartenhorst [3] considered a multi-server Markov queueing model with an infinite queue capacity and server breakdowns as well, but with a limited repair capacity. Some experimental outcomes related to a behavior of an unreliable multi-server Markov queueing system were presented by Mitrani and King in paper [4]. Queueing systems with two unreliable servers were studied by Madan, Abu-Dayyeh and Gharaibeh [5] and by Yue D., Yue W., Yu and Tian [6]. In paper [5] the authors assumed homogeneous servers, in paper [6] heterogeneous servers were considered. Wang and Chang [7] studied a finite Markov queueing system with balking, reneging and server breakdowns.

Aplimat – Journal of Applied Mathematics 2

Assumptions of the model

Let us consider a finite Markov multi-server queueing system consisting of n homogenous parallel placed servers subject to breakdowns. Incoming customers can wait for the service in a finite queue with a capacity equal to m-n, customers are served one by one according to FIFO (First In – First Out) discipline. Thus there are in total m places in the queueing system. Customers come to the system according to the Poisson process with the parameter λ, hence the 1 customers inter-arrival times are exponentially distributed with the mean value equal to . The



customers service times follow the exponential distribution with parameter μ, thus the mean service 1 time is equal to .



Each of the servers is successively failure-free and broken; let us assume that failures of the servers are mutually independent. Furthermore let us consider that a server breakdown can occur at any time. That means the server can break if it is busy or idle. Time of failure-free state is an 1 exponential random variable with the parameter η, thus the mean value is equal to . The server



repairing time is an exponential random variable as well, but with the parameter ξ; the mean 1 repairing time is therefore equal to . Let us assume an ample repair capacity – repairing of each



server starts immediately after the occurrence of the breakdown. And finally let us assume the so called preemptive-repeat discipline that means the customer servicing is interrupted after the occurrence of the server breakdown (if the server is servicing a customer at this moment) and its service starts from the beginning. Generally one of the three different events can occur:  If there is an idle server in the system, the customer will go over to the idle server.  If there is not an idle server in the system, but there is a free queue place, the customer will come back to the queue.  If there is neither an idle server nor a free queue place, the customer will be rejected. 3

Mathematical model

Let us consider a random variable X(t) being the number of broken servers and a random variable Y(t) being the number of customers finding in the system at the time t. On the basis of the assumptions established in Section 2 it is clear that {X(t), Y(t)} is a two-dimensional Markov process with the state space   i, j , i  0,1,..., n; j  0,1,..., m  i. The system is found in the state (i,j) at the time t if X(t) = i and Y(t) = j, let us denote a corresponding probability Pi, j  t  . Let us illustrate the queueing model graphically as a state transition diagram (see in fig. 1). The vertices represent the particular states of the system and oriented edges indicate the possible transitions with the corresponding rate.

198   

volume 4 (2011), number1

Aplimat – Journal of Applied Mathematics

0,0 ξ

μ

jμ

λ

λ

2ξ

iξ

ξ μ

jμ

λ

λ

(n‐i+1)η

i,0 (n‐i)η

(n‐1)ξ

2η

n‐1,0

(n‐1)η

iξ

(n‐i+1)η

λ (i+1)ξ

(n‐1)ξ μ

i,j

nξ

λ

(n‐i)η

λ

2ξ

i,n‐i

λ

λ

λ

0,m nη

1,m‐1 (n‐1)η

(n‐i+1)η iξ

λ

λ

(n‐i)η

(n‐1)ξ

(n‐i+1)η

i,m‐i (n‐i)η

2η 2η

μ

n‐1,m‐n+1

λ η

η

n,j

nη

(n‐i)μ (n‐i)μ

(i+1)ξ

λ

λ

(n‐1)η

i,j

λ

(n‐i)η

λ

(n‐i+1)η

iξ

η nξ

1,j

λ

2η

n‐1,j

λ

(n‐1)μ (n‐1)μ

(n‐i)μ (n‐i)μ

λ

λ

nη

(n‐1)μ

λ

μ λ

ξ

(n‐i+1)η

iξ

μ

η

n,0

nη

(n‐1)η

(i+1)ξ

(n‐1)ξ

2η

n‐1,1

1,n‐1

λ

0,j

λ

λ

nμ

ξ

(j+1)μ (n‐i)μ

μ

λ

ξ

nμ

nμ

(n‐1)μ

2ξ

2ξ

0,n

nη

(n‐1)μ (j+1)μ λ

nμ λ

ξ

nη

jμ

λ

(i+1)ξ

λ

1,j

(n‐1)η

μ

nξ

0,j

nη

1,0

nμ

(j+1)μ

nξ λ

λ

η

n,m‐n

Figure 1: The state transition diagram. On the basis of the state transition diagram we can obtain a finite system of the differential equations for the probabilities Pi, j  t  depending on the time t. For t   we get the system of the linear equations for steady state probabilities Pi , j  that are not dependent on the time t. The steady state balance equations are:

  n P0,0   P0,1  P1,0  ,

(1)

  n  i   i Pi ,0   Pi ,1  n  i  1Pi1,0   i  1Pi1,0  for i  1,..., n  1 ,

(2)

  n Pn,0   Pn1,0  ,

(3)

  j  n P0, j   P0, j 1   j  1P0, j 1  P1, j  for

j  1,..., n  1 ,

(4)

  j  n  i   i Pi , j   Pi , j 1   j  1Pi , j 1  n  i  1Pi1, j    i  1Pi 1, j  for i  1,..., n  2; j  1,..., n  i  1 ,

(5)

  n  n P0, j   P0, j 1  nP0, j 1  P1, j 

volume 4 (2011), number 1 

for j  n,..., m  1 ,

(6)

199

Aplimat – Journal of Applied Mathematics

  n  i   n  i   i Pi , j   Pi , j 1  n  i Pi , j 1  n  i  1Pi1, j    i  1Pi 1, j  for i  1,..., n  1; j  n  i,..., m  i  1 , n  n P0,m   P0,m1 ,

(7) (8)

n  i   n  i   i Pi , m i   Pi , m i 1  n  i  1Pi 1, m i   n  i  1Pi 1, m i 1 for i  1,..., n  1 ,

  n Pn, j   Pn, j 1  Pn1, j  for

(9) j  1,..., m  n  1 ,

nPn ,mn   Pn ,mn1  Pn1,mn   Pn1,mn1 .

(10) (11)

Clearly, the probabilities Pi , j  must satisfy the normalization equation: n m i

 P i 0 j 0

i, j 

 1.

(12)

By solving of the linear equations system, obtained from equations (1) – (10) and (12), we get steady state probabilities of the particular states of the system that are needed for performance measures computing of the studied queueing system. Please notice that the equation (11) is a linear combination of the equations (1) – (10), therefore it is dropped and replaced by the normalization equation (12). On the basis of the steady state probabilities the following performance measures can be computed. The mean number of the costumers in the service, denoted as ES, can be computed according to the formula: n 1 n  i

n 1

i  0 j 1

i 0

ES   jPi , j    n  i 

m i

 P

j  n  i 1

i, j 

.

(13)

The utilization of the servers denoted as χs is expressed by the formula: n 1 n  i

n 1

 jPi, j    n  i 

m i

 Pi, j  ES i  0 j 1 i 0 j  n  i 1  s  . n n The mean number of the waiting customers EL can be expressed by the formula: n

EL  

m i

  j  n  i P

i  0 j  n  i 1

i, j 

.

(14)

(15)

Clearly for the mean number of the customers finding in the system EK it must be satisfied: n m i

EK  ES  EL   jPi , j  .

(16)

i  0 j 1

For the mean number of broken servers EP it can be written: n

m i

i 1

j 0

EP   i  Pi , j  .

(17)

And finally the utilization of the repair capacity χr can be expressed by the formula: 200   

volume 4 (2011), number1

Aplimat – Journal of Applied Mathematics

r 

4

EP  n

n

m i

i 1

j 0

 i  Pi , j  n

.

(18)

Simulation model

In order to validate the outcomes, which were reached by solution of the above-mentioned mathematical model, Petri net model of the studied queueing system was created by using CPN Tools – Version 2.2.0. The software CPN Tools is designed for editing, simulating and analyzing coloured Petri nets. The created simulation model is shown in fig. 2. The model is compound of 11 places and 9 transitions. The Petri net presented in fig. 2 models the unreliable M/M/3/6 (thus n=3, m=6) queueing system fulfilling the conditions mentioned in Section 2, where λ=6 h-1, μ=2 h-1, η=150-1 h-1 and ξ=0,2 h-1. The concrete values of the random variables are generated during the simulation through the defined function fun ET(EX) = round(exponential(1.0/EX)). We apply a minute as the unit of time and the parameter of the defined function is equal to the mean value of the exponential distribution. 1 So the mean value of each random variable must be expressed in [min]. We substitute  10 min,



1



 30 min,

1



 9000 min and

1



 300 min in this case.

Figure 2: The simulation model created by using CPN Tools – Version 2.2.0. volume 4 (2011), number 1 

201

Aplimat – Journal of Applied Mathematics

The created Petri net works with following tokens:  Tokens C represent incoming and waiting customers.  Tokens (C,i), where i  1,2,3, represent customers as well, but in the phase of the service process (each customer in the service is labeled by a number of the server where is serviced); these tokens are used for modelling of available and working servers as well.  Tokens (F,i), where i  1,2,3, model failures of the servers.  Auxiliary tokens P serve for modelling of the free queue places and unavailable servers. The place “Customer” with the initial marking C and the transition “Customer initialization” model the Poisson arrival process of customers; first customer comes to the system at the time 0. A customer, approaching to the queueing system finding in the place “Entry place”, will come in the queue modelled by the place “Queue” if there is a token P in the place “Free places” (the initial marking of the place is equal to the queue capacity – 3`P), otherwise the customer will be rejected through the transition “Rejection”. The transition “Rejection” will be enabled only if there are exactly 3 customers in the queue and all of the servers are unavailable (busy or broken) – the transition is connected with the place “Queue” by the testing arc with the arc expression 3`C and with the place “Unavailable servers” by the testing arc with the arc expression 3`P. The transition “Service beginning” will be enabled if there is a customer in the queue and a server is available (idle) – the place “Available servers” with the initial marking 1`(C,1)++1`(C,2)++1`(C,3) models available servers. In the place “Servicing” there are placed tokens modelling servicing of customers, the appropriate service time is ensured by the time stamp update through the transition “Service beginning” firing. The transition “Service termination” will be enabled if there is a customer in the service and corresponding server is working (or busy). In the place “Breakdowns” with the initial marking 1`(F,1)@+ET(9000.0)++ 1`(F,2)@+ET(9000.0)++1`(F,3)@+ET(9000.0) there are placed tokens modelling failures of servers. When the actual simular time equals to the value of the time stamp of the token modelling breakdown of i-th server, its failure happens. If i-th server is idle at this moment (there is the token (C,i) in the place “Available servers”), the breakdown of the idle server will occur. If i-th server is busy at this moment (there is the corresponding token in the place “Working servers”), the breakdown of the busy server will occur; this breakdown will cause the rejection of corresponding customer through the transition “Rejection due to failure” firing and the input of a new costumer to the place “Entry place”. The tokens (F,i) finding in the place “Repairing” model repairing of servers, the exponential repair time is ensured by the time stamp update. The repairing of i-th broken server is terminated by the transition “Repair termination” firing and corresponding token (F,i) is sent with the updated time stamp back to the place “Breakdowns”. There are three marking size monitoring functions that were applied for the computing of selected performance measures during simulation:  The monitoring function which is bonded with the place “Working servers” enables the estimation of the mean number of the customers in the service ES.  The monitoring function which is bonded with the place “Queue” serves to the estimation of the mean number of the waiting customers EL.  The monitoring function which is associated with the place “Repairing” enables the estimation of the mean number of the servers in failure EP.

202   

volume 4 (2011), number1

Aplimat – Journal of Applied Mathematics 5

Evaluation of executed experiments

Let us consider the studied queuing system with 3 parallel servers (n=3) and in total 6 places in the system (m=6), therefore the queue capacity is equal to 3. Let us consider applied values of the random variables parameters being λ=6 h-1, μ=2 h-1,   1501 ,1251 ,1001 ,751 ,501 h-1 and ξ=0,2 h-1, consequently we consider 5 model configurations differing in the parameter η. The steady state probabilities were obtained by the solution of the linear equations system introduced in Section 3, the equations system was solved numerically by using the software Matlab. On the basis of steady state probabilities knowledge we can compute the performance measures according to the formulas (13) up to (18). Let us focus on the performance measures ES, EL and EP. The experimental estimation of the performance measures we got by simulation of the coloured Petri net presented in Section 4. Thirty independent experiments were executed for each model configuration; each experiment was terminated after a million steps (a step corresponds to a transition firing). With the view of the further statistical evaluation the normality of the simulation outcomes was tested by using the  2 goodness-of-fit test. As for all cases p-value is greater than 0,05 we do not reject individual hypotheses about the normal distribution. We can compute 95% confidence intervals for selected performance measures, because the normality was not rejected. The reached outcomes are summarized in the table 1 below. Please notice that all statistical computations were executed by using software Statgraphics plus 5.0. Table 1: The reached outcomes. ‐1

η  [h ] ‐1

150

‐1

125

‐1

100

‐1

75

‐1

50

Normality testing Performance  measure Mean value μ Dispersion σ 2 ES 2,43403 2,49E‐05 EL 1,09976 3,11E‐05 EP 0,09565 8,40E‐06 ES 2,42187 2,25E‐05 EL 1,11362 3,54E‐05 EP 0,11491 3,03E‐05 ES 2,40414 2,25E‐05 EL 1,13288 3,86E‐05 EP 0,14210 2,73E‐05 ES 2,37679 2,31E‐05 EL 1,16837 3,91E‐05 EP 0,18589 3,33E‐05 ES 2,32002 4,06E‐05 EL 1,23248 3,05E‐05 EP 0,27296 5,51E‐05

P‐value 0,61596 0,24144 0,11569 0,24144 0,85761 0,91608 0,11569 0,70293 0,70293 0,70293 0,36904 0,52892 0,61596 0,95798 0,61596

Analytic  result

95% confidence  interval

Difference [%]

2,43097 1,09160 0,09677 2,41931 1,10555 0,11538 2,40199 1,12614 0,14286 2,37355 1,15954 0,18750 2,31825 1,22311 0,27273

(2,43216; 2,43589) (1,09768; 1,10184) (0,09457; 0,09673) (2,42009; 2,42364) (1,11140; 1,11584) (0,11286; 0,11697) (2,40236; 2,40591) (1,13056; 1,13520) (0,14015; 0,14405) (2,37499; 2,37858) (1,16603; 1,17070) (0,18373; 0,18804) (2,31764; 2,32240) (1,23041; 1,23454) (0,27019; 0,27573)

0,13 0,75 ‐1,16 0,11 0,73 ‐0,41 0,09 0,60 ‐0,53 0,14 0,76 ‐0,86 0,08 0,77 0,09

On the basis of the reached outcomes it is possible to say that there are statistical significant differences for all the cases, because the analytic results do not fall into the corresponding confidence intervals. However these differences are not considerably in practice as we can see in the last column of table 1. In this column the percentage differences are shown between the analytic volume 4 (2011), number 1 

203

Aplimat – Journal of Applied Mathematics

results and the estimations of the performance measures mean values gained by the simulation experiments Let us illustrate the gained relations graphically – see figures 3, 4 and 5. In all graphs two lines are drawn, first line denoted as “Analytic result” corresponds to the value gained by the mathematical model solution and the second line “Mean value” to the estimation of the mean value (sample average) gained by the simulation experiments. The figure 3 shows the relationship between the mean number of the customers in the service ES and the mean time of failure-free state 1 1 . We can see that the increasing value causes the increasing of ES, this relationship could be





logically expected. 2,44 2,42

ES

2,40 2,38 2,36

Analytic result

2,34

Mean value

2,32 2,30 50

75

100

125

150

Mean time of failure‐free state [h]

Figure 3: The relationship between ES and

1



.

1,24 1,22 1,20

EL

1,18 1,16 Analytic result

1,14

Mean value

1,12 1,10 1,08 50

75

100

125

150

Mean time of failure‐free state [h]

Figure 4: The relationship between EL and

204   

1



.

volume 4 (2011), number1

Aplimat – Journal of Applied Mathematics

The relation between the mean number of the waiting customers EL and the mean time of failure1 free state is shown in figure 4. And finally the dependence of the mean number of broken servers



EP on the mean time of failure-free state

1



is drawn in figure 5. Both relationships are logically

decreasing. 0,30 0,25

EP

0,20 Analytic result

0,15

Mean value 0,10 0,05 50

75

100

125

150

Mean time of failure‐free state [h]

Figure 5: The relationship between EP and

5

1



.

Conclusion

There were presented two models of the M/M/n/m queueing system with the servers subject to breakdowns and an ample repair capacity in this paper – the mathematical model and the simulation model created as coloured Petri net by using CPN Tools. The major part of the paper was focused on the mathematical model of the studied system and on the description of the created simulation model. At the end of the paper reached outcomes were evaluated for the unreliable M/M/3/6 queueing system with the concrete parameters of the random variables. Please notice that more accurate simulation outcomes we would probably obtain by using lower unit of time, for example a second. As regards to following research on the studied queueing system we would like to find the formula for the customer rejection probability above all. And finally, the model will be generalized by the assumption of an insufficient repair capacity r