ANALYSIS OF MULTI-SERVER SINGLE QUEUE

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with Multiple Phases

Ekpenyong, Emmanuel John

Department of Mathematics, Statistics and Computer Science Michael Okpara University of Agriculture, Umudike, Nigeria [email protected]

Udoh, Nse Sunday

Department of Mathematics and Statistics University of Uyo, Nigeria

[email protected]

Abstract

This paper extends and improves on the performance measures of the Single-Server Single Queue System with Multiple Phases. The extension results in a new Queuing System of Multi-Server with Multiple Phases under the conditions of First Come First Served, infinite population source, Poisson arrivals and Erlang service time. Queuing properties such as expected total service time, its variance and some performance measures like the expected number of phases in the system, expected number of phases in the queue, expected number of customers in the queue, expected waiting times in the queue and in the system as well as the number of customers in the system have been derived in this work for this Multi-Server with Multiple Phases (M/Ek/s: (/FCFS)) queuing model with k identified stages in series. These performance measures so obtained were compared with those of the already existing Single-Server with Multiple Phases (M/Ek/1: (/FCFS) model. Numerical illustration indicates the efficiency and effectiveness of the latter over the former.

Keywords: Multi-server, Single queue, Multiple phase queue model, Erlang distribution, Performance measures.

1.0 Introduction

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Ekpenyong, Emmanuel John, Udoh, Nse Sunday

and Erlang service time (M/Ek/s:(/FCFS)), which is an extension of the M/Ek/1:(/FCFS) model with multiple service channels; see Sharma (2007) and Gupta (2008).

Maurya (2009) analyzed multiple service channels queuing with practical situations arising at those places where phase service is provided under priority queue discipline. He added that the models become more significant when the source of input customers is to be classified into two or more categories. Dimitriou and Langaris (2010) analyze a repairable queuing model with a two-phase service in succession provided by a single server. They considered a case where customers arrive in a single queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed time. Luh (2004) considered a queuing model of general servers in tandem with finite buffer capacities. He studied the probability of blocking and in order to obtain the steady state probability distribution of this model, he constructed an embedded Markov chain at the departure points. He further solved the model analytically and its analysis is extended to semi-Markovian representation of performance measures in queuing networks.

2.0 Method

The Model: The model is made up of ‘s’ multiple service channel with k identical stages in series, each with average service time of _

s

1 _{. The distribution of total}

servicing time of customers in the system is some joint distribution of time in all these stages.

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Figure 1.0: Model of single queue with S servers in k-phases

3.0 Determination of Expected Total Service Time and its Variance

Let s denote the number of customers served per unit time, then ks will be the number of phases served per unit time. The probability density function for Erlang distribution is;

0 ,

)! 1 (

) ( )

( 1 _



 _t  _e _t

k ks t

f  k k kst

dt t f t T

E

o ( )

)

( 





dt e t k ks

t k k ks t

o



 









1

)! 1 (

) (

 s

1 

with variance ₂ ) (

1

 s k

4.0 Derivation of Performance Measures of the M/Ek/s:(/FCFS) Model

Consider Pn(t) to be the probability that there aren customers in the kphases in

the system at time t. Let Pn(tt) be the probability that there are n customers in the system, t be the probability that a customer is in the queue at time t,

t



 the probability that a customer is served and kstis the probability that the customers in the s service points in all the phases. Pn1(t) and Pn1(t) are probabilities that there are n+1 and n-1 customers in the system respectively. The probability that the system will contain n customers at time (t+∆t) in the k-phases can be expressed as the sum of joint probabilities of mutually exclusive and collectively exhaustive cases. The difference equations for the model are given as follows:

 

4.2 ,

) ( ] 1 )[ ( ) (

1 . 4 1 , ) ( ) ( )

( )

( ) (

) (

1

2 1

  o

n t ks t P t t

P t t P

n t t P t t P t t P ks t t P ks t

t P

o o

n k

n n

  

  

 

 

 

 

  

 _ _









Arrivals

λ =arrival rate

1, 2 . . . N Queue

1

2

S

1

2

S

1

2

S

…k

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Ekpenyong, Emmanuel John, Udoh, Nse Sunday

The steady state system of difference equations are obtained as:

 

4.4 ... 0 ; 0 3 . 4 ... 0 ; 0 ) ( 1 0 1           _ _ n ks n ks

ks n n n k

      







ks

Let  _{and dividing (4.3) and (4.4) by} ks, we have:

) 6 . 4 ( 0 , ) 5 . 4 ( 1 ; ) 1 ( 0 1 1       _ _ n n n k n n      

To solve (4.5) and (4.6), the method of generating function is used:

Let ( ) ;/ / 1

 

4.7

0  



  y y y G n n n 

Multiplying (4.5) byyn_{and summing over the range, we obtain}

 

4.8 )

1 (

1 1

1



           n n n n n k n n n

ny   y  y

 

An addition of 0to LHS of (4.8) and 1 0to RHS of (4.8), yields:

 

1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 0 1 1 ) 1 ( ) 1 ( 9 . 4 ) 1 (                         



           _          _        n n n n k n k n n n n n n n n k n n n n n n n n n k n n n n n y y y y y y y y y y





























Since _n__k 0for nk 0, we have

      _          



           0 0 0 1 0 0 0 1 1 ; , 1 ) 1 (           i i i m m m k i i i k m m m n n n y y y y i n m k n y y y y ] ) ( [ 1 ) ( ) ( ) 1

(  G y 0  y G y  _y G y 0

or k

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 



_k



n n

n _y _y _y

y y

G

Therefore 2 1

0

0 1 ...

,        







_k



n

n _y_ _y _ _ _y





  ... 2 0 0   (4.11)

Settingy=1 in (4.11),the terms in the bracket would be k k

i k _



1

1 , therefore

 



          0 0

0 ₁ 1

1 n n n k k G     (4.12)



     0 1 ) 1 ( ) 7 . 4 ( , 1 n n G gives y

For  (4.13)

 



 or k

k _  

      1 1 1

1 0 0 (4.14)

Putting the value of 0 in (4.10), we obtain



4.15



) 1 ( ) 1 ( ) ( ) 1 ( ) ( 0 n n k n

n _y _y

y k

y

G  

     











  



                          _                        _  

0 0 0

1 0 2 0 1 0 0 16 . 4 ) 1 ( ) 1 ( 1 ) 1 ( ; ) 1 ( ) ( ) 1 (

n n z

n zk i i i n z i n z n n n i i i i i n zk z z n z n n y C C k y y C y C y k







Comparing the coefficient of yn_{in both sides of (4.16), we obtain}



4.17



) 1 ( ) 1 ( 1 , , i i n z n z i z n n

n  k



  C  C



5.0 Results

To determine the expected number of phases in the system , ) ( 0



   n n s k n

L  we consider one of the difference equations as: 1

; )

1

(  n nk n1 n

Multiplying both sides byn2_{and summing gives:}

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If nk nand n1n,then RHS becomes;





0

0 2 2 0 0 2 0 2 1 2 0 2 1 2 ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 (                                 



            n n n n n n n n n n n n n k n n k n n k n n





0 0 2 0 0 2 1 2 0 0 2 2 2 ) 1 ( 2 ) 1 ( 2 ) 1 ( ) 1 ( ) 2 1 ( ) 2 (                           



          n n n n n n n n n n k n k n n n n nk k n

 

5.1 1 , , 2 ) 1 ( ) ( ) ( 2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 1 1 ) ( L ) 1 ( 2 ) 1 ( 1 ) 1 ( 2 1 ) ( L 1 ) 1 ( 2 , 2 s 2 0 2 0 0 s 0 2 0                                          



      k and s ks s k s s ks k k k k k k k k k k k k k k k k pk n n k k n k hence n n n n n n                                

The other operating characteristics of the model have been obtained as follows:

Expected Number of Phases in the Queue

 

5.2 ) ( 2 1 ) ( L ) ( L s q                









s s

k s

k k

Expected Number of customers in the Queue

k Service in Phases of Number Average k 

 L ( )

L s

q

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Expected Waiting Time in the Queue

 

5.4 )

( 2

1 Lq

   

 

 

       







k s s

k Wq

Expected Waiting Time of a customer in the system

 

5.5 1

) (

2 1

1 2











k s s s

k s

W

Ws q 

  

 

 

        

Expected Number of customers in the system

 

5.6 L

1 ) (

2 1 L

L

s

2 q

s

W or

s s

s k k s

















    

 

 

        

6.0 Applications Illustration 1

In a National Youth Service Camp in Nigeria, there are four counters through which all Corp members must pass before Registration and accreditation are completed. At the first counter, registration is done; in the second counter, there is a presentation of credentials for authentication. At the third counter, traveling allowances are paid and at the fourth counter, assignment of Corp members to various hostels is done. If arrival of Corp members follows a Poisson process with a mean arrival of 9/hr, the service times follow an Erlang distribution with a mean of 1.5 minutes per Corp member, and the Queue discipline is first come first served with infinite population; the performance measures can be obtained as follows:

CASE 1:

We note that this is a case of Multiple Phases Queue model with a single server in each phase, that is,M/EK/1

λ = 9/hr, µ= 1.5mins/Corp member,=10 Corp members/hr

k = 4

The following results are obtained:

members Corp

k

k ₅_.₀₆₂₅ ₅

) 9 10 ( 10

9 8

5 ) ( 2

1

Lq 2 2   

  

 

 

   

 

 









,

s hrs

k 1 5 9 ₀_.₅₆₂₅ ₃₃_.₇₅_min

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Ekpenyong, Emmanuel John, Udoh, Nse Sunday

, min 75 . 39 6625

. 0 10

1 5625 . 0 1

Ws  Wq _    hrs  s

members Corp

Ws 9 0.6625 5.9625 6

Ls      ,

and E T 0.1hror6mins

10 1 )

(  

Case 2:

Now, suppose each counter has 2 service points, we can obtain the performance measures as is the case in this work usingM/EK/2

From (5.3), 0.023

17600 405 )

9 20 ( 20

9 8

5

Lq 2   

  

 



 ;

(5.4) gives 0.00256hrs 0.153mins 9s

17600 9 5

Wq   



 ;

(5.5) is hr s

s

Wq 1 0.00256 ₂₀1 0.05256 3.15min

Ws  



    ;

Also, (5.6) gives Ls Ws  90.05256  0.473

and hror s

s T

E 0.05 3min

20 1 1 )

(   



In contrast, the performance measures in case 2 give improved results in handling congestion through queuing model. For instance, in case 2, Wq = 9S indicates that in adopting the two-server multi-phase queuing model, a Corp member can only wait in the queue for 9 seconds; the result Lq = 0.023 shows

that by adopting this model there will be little or no congestion at all. Whereas in case 1, a Corp member can spend 33.75mins in a queue before he is being served.

Illustration 2

Suppose the Corp members arrive at the mean rate of 18 per hour while the mean service time is still 1.5mins per Corp member. We find out in this case, that the application of case 1 will be impracticable, because arrival out-matches the service rate, such that

0 )

( 2

1

Lq 2  

  

 

 

      







k k

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On the other hand, case 2 which is Multiple-Server Multi-Phase Queuing System becomes more practicable, since it involves more service point at each phase. Thus:

(5.3) gives Corpmembers

s s k

k ₅_.₀₆₂₅ ₅

) 18 20 ( 20

) 18 ( 8 5 ) ( 2

1

Lq 2 2   

  

 

 

   

 

 



  

 _;

(5.4) gives 0.281hr 16.875mins

18 0625 . 5 L

W q

q  _    ;

(5.5) is hr s

s

Wq 0.33125 19.875min

20 1 281 . 0 1

Ws  



    ;

and (5.6) gives Ls Ws  180.331255.9625  6Corpmembers.

Also, hror s

s T

E 0.05 3min

20 1 1 )

(   



7.0 Conclusion

In this paper, we have extended the M/Ek/1: (/FCFS) model with single server

to M/Ek/s:(/FCFS) model with multiple server and obtained its performance

characteristics which is an improvement of the M/EK/1: (/FCFS) model for

handling congestion especially at peak period which in turn sustains customer’s goodwill. Numerical illustrations have shown that the extended M/Ek/s: (/FCFS)

model outperformed the single server M/EK/1: (/FCFS) model when the arrival

rate of customers is very high with a fixed mean service time. In the first illustrations, it is observed that with M/Ek/1: (∞/FCFS), an average of five corps members would wait in the queue for service and each corp member would spend 33.75 mins waiting for service. In contrast, the use of M/Ek/s: (/FCFS)

model indicates that little or no queue would be obtained and a corp member can only wait for 9seconds only for service. These results show the efficiency and effectiveness of the latter model over the former. Further, in the second illustration, case 1 cannot be applied here because the traffic intensity here could not contain this model. Therefore the second case becomes suitable and practicable.

References

1. Dimitriou, I. and Langaris, C. (2010). A Repairable Queueing Model with Two-phase Service, Start-up Times and Retrial Custom, Journal of

Computers and Operations Research, 37(7), 1181-1190.

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3. Luh, H. (2004). A Queuing Model of General Servers in Tandem with Finite Buffer Capacities. International Journal of Operations Research, 1(1), 71-76.

4. Maurya, V. N. (2009). Analysis of a Multiple Service Channel Queuing Model with Priority Queue Discipline and Phase Service-Part I. International J. of Maths. Sc. & Eng. Appls. (IJMSEA), 3 (IV), 357-364. 5. Reid, R. D and Sanders, N. R. (2010): Operations Management. Wiley

and Sons. 3rd_{Edition. New York.}