IRJET- Two-Class Priority Queueing System with Restricted Number of Priority Customers and Catastrophes

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2004
Two-Class Priority Queueing System with Restricted Number of
Priority Customers and Catastrophes
A.M.K. Tarabia1, A.H. El-Baz2, A.M. Elghafod3
1Professor of Mathematical Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt.
2Assistant Professor of Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt .
3M.Sc. School Basic Sciences – Department of Mathematical Sciences –Statistics Division - New Damietta, Egypt.
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Abstract - In this paper, we analyze a two-class single-server preemptive priority queueing system with restricted number of
priority customers and catastrophes. The arrivals per class follow the Poisson processwithexponentiallydistributedservicetimes.
Customers are served on a first-come, first-served basis within their queue. Explicit expressions for the mean queuelengthandthe
joint distribution are derived in the steady-state case for the number of high and low prioritycustomersinthesystem. Theanalysis
is based on the generating function technique. Also, we study the impact of catastrophe on the system and other performance
measures.
Key Words: Queueing, Queue length, Preemptive resume, Generating function, Catastrophe
1. INTRODUCTION
This Priority queues occur in many aspects of daily life, especially in cases where preferential treatment is given to certain
types of individuals, for example. Telecommunications field. Priority mechanisms are an invaluable scheduling method that
allows messages from different categories to receive different quality of service. For this reason, the priority queue has
received considerable attention in the literature. Cobham [1] wasthefirsttoconsiderthenon-preemptivepriority ofthequeue
was considered with Poisson inputs and exponential retention time and discussed them for single and multi-channel cases.
Holley [2] simplified Cobham’s work. Barry [3] and Stephan [4] studied the problem of preemptive priority system problem
with two priorities, Poisson inputs and exponential retention time. White and Christie [5] and Heathcote [6] studied the same
problem for two channels and used the generating function method to describe the steady-state and transient queue length
distributions, respectively. The main difficulty is to extend these results to multiple categories with general service time
allocations in processing Laplace transformations that characterize the dynamics of the system. An extensive study of two
priority plans can be cited in Miller [7] where he obtained several results in this model using the functions of generating and
transformations of Laplace. A different approach to converting Laguerre to this model has also been applied, see for example
Keilson and Sumita [8] and Keilson and Nunn [9]. Miller [10] considered exponential single server priority queues with two
classes of customers. He Obtain repeated calculation formulas for steady distributions using Neuts [11] theory of matrix-
geometric invariant probability vectors. Shack and Larson [12] are treated with a multi-server, non-preemptive, Multi-
stationary queueing system with Poisson arrivals and exponential servers. They assumed a limited discipline in the priority
queue and derived many performance measures. Franti [13,14]developedalgorithms fora dynamicpriority queue tocompute
performance measures such as the distribution of queue length and moments. Nishida [15]gaveanapproximation analysisfor
a heterogeneous multiprocessor system with preemptive and non-preemptive priority discipline and also presented the
numerical comparison between simulation and approximation analysis. Recently, Bitran and Caldentey [16] analyzed a two-
class single queueing system that features state dependent arrivals and preemptivepriorityservicediscipline.Forthissystem,
they computed the mean balance queue length and studied the effective servicetimeandotherfirstpassagetimequantities for
both classes. More recently, Drekic and Woolford [17] analyzed a two class, single-server preemptivepriority queueingmodel
with low priority balking customers. They considered arrivals to each class are assumed to follow a Poisson with
exponentioally distributed service times and obtained the steady state jointdistributionofthenumberofhighand lowpriority
customers in the system. Choa and Zheng [20] considered an immigration birth and death process with total catastrophesand
studied its transient as well as equilibrium behavior Kirshna Kumar et al. [23,24] obtained transient solution for analytically
using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and nonzero
repair time , the steady state probability of the system size, and obtained transient solution for the system size in the M/M/1
queueing model with the possibility of catastrophes at the service station in the direct way. Dharmaraja and Rakesh

Kumar [25] used the generation function technique to derive the model's transient solution directly. Explicit time dependent
probabilities are obtained for system size . Other contributions are due to Economou and Fakinos [26,27] that extend some
previous results on birth-death processes with catastrophes to the more general cases of continuous-timeMarkovchainswith
catastrophes resulting from a point process, and of the non-homogeneousPoissonprocesswithtotal orbinomial catastrophes.
Following this introduction, the present is organized as follows. In section 2, we describe themodel alongwiththeassumption
and several notation used throughout the paper. Moreover, we obtain the generating function for queue size of each priority
classes. Using the obtained generating function and partial fractions technique, we derive explicit expressions for the steady-
state probabilities. Finding explicit expressions for ,.iP and ., jP themarginal distributionsofthehighandlowpriorityclasses,
respectively is presented in Section 3. In Sections 4 include a simple way to obtain and compute the mean queue length of the
high priority classes. In Section 5, several special cases obtained to support our work and conclusion. Also, some numerical
results conclude the paper in Section 6.
2. Model Description and Analysis
We consider a single server queueing system servingtwotypesofcustomers;class-1andclass-2eachhavingitsownrespective
line and the arrival process for both types is state independent. A higher priority is assigned to class-1.Supposethattheruleof
service within each class is FCFS and that the priority system is preemptive resumed. i.e. during the service of a low priority
customer, if high priority customers joins the system, then the low priority customer’s service will stop and will resume
working when there are no high-priority customers in the system. Let also by the number of the customers of class i (i = 1, 2).
We denote the number of customers in the first class is restricted to a finite number L including the one being served, if any,
and the number of the second class is infinite. Suppose also 1 2,  denote the arrival rates for the two classes and let 1 2, 
denote the service rates for the two classes. We definethe trafficintensitiesby 1 2
1 2
1 2
,
 
 
 
  andthe steadystateprobability
that the system is in state (i, j), where i is the number of the high priority customers and j is the number of low priority
customers in the system. The catastrophes also  Whenever a catastrophe occurs at the system, all the available customers
there are destroyed immediately, the server gets inactivated momentarily and the server is ready for service when a new
arrival occurs. Clearly, the governing difference equations of the system under consideration are given by
1 2 0,0 1 1,0 2 0,1( )P P P          (1)
1 2 2 0, 2 0, 1 1 1, 2 0, 1( ) , 1j j j j jP P P P             (2)
1 2 1 ,0 1 1,0 1 1,0 , 1 1( ) i i iP P P i L            (3)
1 2 1 , 1 1, 2 , 1 1 1,( ) , 1 1, 1i j i j i j i jP P P P i L j                  (4)
2 1 ,0 1 1,0( ) L LP P       (5)
2 1 , 1 1, 2 , 1( ) , 1L j L j L jP P P j          . (6)

Then the above system (1) – (6) can be written as
1 1
21 0 2 0,02 1 1( () (1 ) ( ) (1 ))G ss G s s Ps     
        (7)
 1 1 1 1 2 1 1( ) ( 1) ( ) ( ) 0, 1 1i i iG s s G s G s i L                 (8)
1 1 1 2( ) ( ( 1)) ( ) 0L LG s s G s         , (9)
where ( )iG s is the generating function for the steady state probabilities ,i jP . Now, these equations can be written as
( ) ( ) ( )A s G s B s , (10)
where  0 1 2( ) ( ) ( ) ( ) ..... ( )
t
i LG s G s G s G s G s is the generating function state probability vector for ,i jP ,
1
2 0,0( ) (1 ) 0 0 .... 0
t
B s s P  
     and ,( ) i jA s a    is ( 1) ( 1)L L   order tridiagonal matrix and its
coefficients are given by
1
1 2 2
1
, 1 , 1 2
1
( )(1 ), 0
, 1, 0,1,2,...,
(1 ) ( 1), , 1,2,....,
, 1, 1,2,....,
0, .
i j i L
s s i j
j i L
a s j i i L
j i i L
otherwise
   

    

      

  


       

   



We find explicit expression for ( )iG s by the following theorem.
Theorem 1. For any non-negative integer , 0i i L  , we have
1
1 2 0,0
2 1 2 1 , 1 1 2
0
( (1 ) ) ( )
( ) ,
( ( 1)) ( )
i
L i
i L
L k
k
s P D s
G s
s s
  
         



 

      
(11)
where
1 1 1 2( ) ( ) ( ) 2 ,m m mD s D s D s m L       (12)
with

 
1 1 2
2 1 2 1 1
1 1 2
1 1
1
0,0 21 1
2
2
1 1 1 1 1 1
2
1 1 1 1 1 1
'
0
( ) ( 1),
( ) ( 1) ,
( 1),
( ) ( )
( ),
( )
( ) ( ) 4
,
2
( ) ( ) 4
,
2
( ) (1),
L L L L
L L
L
i
i
D s s
D s s
s
a b a b
P E Y
a b
a
b
E y G
  
     
    
 


       
       
 
 

   
    
    
  
  

     

     

 
1 1 1 2 2 2,       and , , 1,2,3,...,L k k L  are the root of Chebychev's polynomials of second kind.
Proof. The proof closely parallels the elementary construction of continuous-time Markovian queue, see for instance
Tarabia [ 19]. We will start by solving above system (10) using Crammer's rule, we obtain
( )
( ) , 0,1,2,3,...,
( )
i
i
A s
G s i L
A s
  , (13)
where ( ) , 0,1,2,...,i
A s i L is determinant of thematrix ( )A s with replacing its ith column bytheelementsofthevector
( )B s . Expanding the determinant ( ) i
A s , we easily obtain
1
1 2 0,0( ) ( (1 ) ) ( ), 0,1,2,...,i
L ii
A s s P D s i L   
    , (14)
where ( )L iD s represents the determinant of thebottomrightsquarematrixobtainedbyeliminatingthefirstrowandcolumn
of the matrix ( )A s and it can be satisfied the following recurrence relations
1 1 1 2( ) ( ) ( ) 2 ,m m mD s D s D s m L       (15)
where
 1 1 2 2 1 2 1 1( ) ( 1), ( ) ( 1) ,D s s D s s                 and 1 1 2 ( 1).s        
On other hand, we do some operations on the matrix ( )A s and it can be seen that satisfies the following recurrence relations
1
1 1 2 1 1 1 2( ) ( ) (1 ) ( ) ( ) ,L LA s s C s C s      
 
          (16)
1 1 1 2( ) ( ) ( ), 2 ,n n nC s C s C s n L       (17)
where 1C  and
2
2 1 1.D    
So

 1
1 2 0,0
1
1 1 2 1 1 1 2
(1 ) ( )
( ) , 0,1,2,...,
( ) (1 ) ( ) ( )
i
L i
i
L L
s P D s
G s i L
s C s C s
  
      



 
 
 
        
(18)
Now, to compute 0,0P we add all the Equations (7) – (9) for 0,1,....j  . then we take limited at 1s  , we get
'
2 2 0
1
0 0
2 2
0 0 0
'
2 0,0 0
'
( ) ( ) ( )
( ) ( ) ( )
lim ( 1) ( )
( 1) ( 1)
L L
s
i i
L L L
i i i
i i
i i i
P G s s G s G s
G s G s G s
s G s
s s s
 
  
 

 
  
  
     
 
 
     

 
 
 
  
'
2 2
0 0 0
2 0,0 0 (1) (1) .(1) ( )
L L L
i i
i i i
i
P G G G G s   
  
 
  
     
  
   (19)
We use the facts
0
(1) 1
L
i
iG

 and
0
'
((1) ),
L
i
i
E yG

 we obtain
0 0,0 2
2
(1) ( ),G P E y



   (20)
Using Equation (18) with i=0,s=1, we get
 0
1 1 1 2
(1)
(1) ,
(1) (1)
L
L L
D
G
C C   


(21)
where
1 1 1
1(1) ( ) [( ) ( )],L L L L
LD a b a b a b  
     (22)
1
1(1) ( ) ( ),L L
LC a b a b
    (23)
1 1 1
2 (1) ( ) ( ).L L
LC a b a b  
    (24)
Substituting from Equations (22)-(24) into Equation (21), the marginal probability 0,.P can be written as
1 1
1
0 0,. 1 1
( ) ( )
(1) ,
( )
L L L L
L L
a b a b
G P
a b
 
 
  
 

(25)
On the other hand, Equations (20) and (25) together with some little simplifications imply that
1 1
1
0,0 2 1 1
2
( ) ( )
( ) .
( )
L L L L
L L
a b a b
P E y
a b



 
 
  
  

Then

1 1
1
0,0 21 1
2
( ) ( )
( ).
( )
L L L L
L L
a b a b
P E y
a b
 


 
 
  
  

(26)
Proposition 1. For any , 0i i L  we have
0 2
, 2 1
12 2( ) ( )
j
L
j k
i j j
k k
A A
P


     

 
  
  
 (27)
Proof. The proof this proposition is directly from the using of partial fractions and using the definition of Chebychev's
polynomial of second kind (see Abramowitz and Stegun [28]) A can be written as
 2 1 2 1 , 1 1 2
0
( 1) ( ),
L
L k
k
A s s         

        (28)
where 1 2 1 , 1 1 , 1,2,..., .k L k k L             Hence
 
0
12 1 2 1 , 1 1 2
( )
( ( 1)) 2
L
k
i
k L k
A A
G s
s s         
 
      

0
12 2
2
2
( ) , 0,1,2,..., .
( ) 1 1
L
k
i
k
k
k
A A
G s i L
s s 
  
  

  
   
     
   
 (28)
Extending in power of s and comparing the coefficient of in the both sides drives to , .i jP
0 2
, 2 1
12 2
,
( )
j
L
j k
i j j
k k
A A
P


     

 
  
  

with
1 1 1 2 2 0,0
0 1
1 2 1 2 1
(1 )( )
,
( )(1 ) (1 )
i
L L
P
A
      
      
  

   
1
1 2 0,0
2
2
0
( (1 ) ) ( )
, .
( ( 1)) ( )
i
k L i k k
k kL
k k i
k
k i
s P D s
A s
s
   

   




 
 
  

3. The Marginal Distributions of the High and Low Priority Classes
Finding explicit expressions for ,.iP and ., jP the marginal distributions of the high and low priority classes, respectively is
obtained in the following propositions
Proposition 2. The marginal distributions of the high priority is given by
1 1
1 1
,. 1 1
( ) ( )
, 0,1,2,...,
( )
i L i L i L i L i
i L L
a b a b
P i L
a b
      
 
     

(29)
Proof. As is clearly defined, (1)iG represents the marginal distribution of high priority, thus onecanputintoEquation(18),we
may get
 
1
1 1 1 2
(1)
(1) , 0,1,2,..., ,
(1) (1)
i
L i
i
L L
D
G i L
C C

  

 
 

where
1 1 1
1(1) ( ) [( ) ( )],L i L i L i L i
L iD a b a b a b      
      (30)
now using Equations (23),(24) and (30), we obtain
1 1
1 1
,. 1 1
( ) ( )
(1) ,
( )( ) ( )
i L i L i L i L i
i i L L L L
a b a b
P G
a b a b a b a b
      
 
     
     
where a b   and 1 1 ,a b   then
1 1
1 1
,. 1 1
( ) ( )
, 0,1,2,...,
( )
i L i L i L i L i
i L L
a b a b
P i L
a b
      
 
     

Proposition 3. The marginal distributions of the low priority is given by
2 2 2 2
., 0, 1 0,2
12 2 2 2 2
, 0
( ) ( ) ( )
j k j
j
j j k
k
P P P j
    
         



   
      
       
 (31)
Proof. By adding Equations. (1), (3) and (5) for , 0i i L  , we get
1 1
2 .,0 1 ,0 1 ,0 1 ,0 1 ,0 2 0,1
0 0 0 0
( ) ,
L L L L
i i i i
i i i i
P P P P P P       
 
   
         
which leads to
2
.,0
2 2
.
( ) ( )
P
 
   
 
 
In general and using similar approach by adding Equations. (2), (4) and (6) for , 0i i L  , we obtain
1 1
2 ., 1 , 1 , 2 0, 1 , 1 , 2 ., 1 2 0, 1
0 0 0 0
( ) ,
L L L L
j i j i j j i j i j j j
i i i i
P P P P P P P P        
 
 
   
          

which simply leads to
2 ., 2 0, 2 ., 1 2 0, 1( ) .j j j jP P P P        
By induction method, the marginal probabilities ., jP satisfy the following recursion relation
2 2 2 2
., 0, 1 0,2
12 2 2 2 2
, 0
( ) ( ) ( )
j k j
j
j j k
k
P P P j
    
         



   
      
       

4. The Mean Queue Length of the High Priority Class
In this section, our main objective is to get the mean queue length of high priority class. To obtain it, we have supposed that
1N be the queue length of the first class and define
1 , ,.
0 0 0
( )
L L
i j i
i j i
E N i P i P

  
  
By substituting from Proposition 2 into the above equation, we have
1 1
1 1
1 1 1
0
1 11 1 1 1
1 11 1
0 0 0 0
1
11 1
0 1
( ) ( )
( )
( )
1
( )
1
( )
( ) ( )
i L i L i L i L iL
L L
i
i i i i
L L L L
L L L L
L L
i i i i
L
L i
L L
i
a b a b
E N i
a b
a i b i a i b i
a b a b a b
d
a
a b ad
a
 
   
 



     
 

 
 
   
 

    

        
           
         





   
 1
1
0 1
2 1 2 11 1
1 1
0 01 1
( )
( )
( ) ( )
( ) ( )
L
L i
i
L L
L i L i
i i
d
b
bd
b
d d
a b
a bd d
a b



 
 
 

 
 




  


 
1
( 1) 1
1 1 1
0 01
1 1 1
1
( )
( ) ( )
L L
k L k k L k L
k k
L L
b a a a b
E N
a b b
  



   
 
 
 
  
 
  
  
 
(32)
5. Special Cases
a) If we take 1a  and 1,b  we obtain
1
1 1 1
1 1
1 1
1 ( 1)
( )
(1 )(1 )
L L
L
L L
E N
  
 


    
 
which agress with the know result given by Tarabia [20]andSharmaandGupta[21]when t   .Moreover,itshows
that high priority queue is functioning independently as an M/M/1/L queue.

b) When there is no catastrophe, i.e., ( 0)  and we put 2 0  , we get all of result the well known M/M/1/L model
such as
1 1
1 11
1
(1 )
( ) , 0,1,2,..., , 1.
(1 )
i
L
E N i L
 

 

  

6. Numerical Results
To conclude, weshall now make use of the results outlined in the foregoing to indicatebymeansofsomeexamplesthegoodness
of our continuous approximation to the queueing system with catastrophes. For different values of parameters 1 1 2 2, , ,   
and , we study the impact of catastrophe  on the steady state probability of the system, when
1 1 2 20.12, 0.2, 0.2, 0.57       and 25L  on 0,0P of the given system and the results are shown in Table1 .
Table -1: Effect of  on the steady state probability of the system 0,0P
when 1 1 2 20.12, 0.2, 0.2, 0.57       and 25L 
 0.02 0.2 0.5 0.6 0.85 0.98
0,0P 0.152632 0.428241 0.584867 0.620027 0.693817 0.727395
Chart -1: Effect of  on the steady state probability of the system 0,0P
From Table (1) and Chart (1) all show that the 0,0P increase as the catastrophe rate increases.
7. CONCLUSIONS
We remark that the explicit expressions obtained for the mean queue length and the steady state joint distribution of the
number of high priority customers in the system are simple and with catastrophe . Moreover, on the whole, the analysis of the
first class behaves like M/M/1/L Markovian queue.
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945–958.
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