The Steady-State Analysis of finite waiting space Queuing System with Multiple Parallel Channels in Series attached to Non-Serial Servers both with Balking and both with Reneging

Meenu Gupta* , Man Singh ** , Deepak Gupta***

*A.P. ,Dept. of Mathematics, Dr.B.R.A.Govt.College,Kaithal-136027(Haryana)– India.

** Ex Prof., CCS, Haryana Agricultural University , Hisar – India.

*** Prof. & Head , Dept. of Mathematics, M.M.Engg.college, Mullana (Ambala)- India

Abstract :

The applicability of such models can be found in various industrial production sectors where it is common practice of using several machines simultaneously and then passing the product to various non serial machines for packing according to weight of the packet. We study the steady-state analysis of the present queuing system in the sense that:

M service channels in series with balking and reneging are linked with N non-serial channels having reneging and balking phenomenon where each of M service channels has identical multiple parallel channels.

The input process is Poisson and the service time distribution is exponential. The service discipline follows SIRO-rule (Service in random order) instead of FIFO-rule

(first in first out). The customer becomes impatient in queue after sometime and may leave the system

without getting service . The input process depends upon the queue size in serial and non-serial channels. Waiting space is finite.

Keywords: Poisson stream, Reneging , Balking, Traffic intensity, Steady-state, Parallel channels.

1. Introduction

The problem of serial queues was studied by O’ Brien (1954), Jackson (1954), Barrer (1955), Hunt(1955) and Maggu (1970) in steady-state with Poisson assumptions with the restrictions that the customer must go through each service channel before leaving the system. Singh (1984) studied the problem of serial queues introducing the concept of reneging. The steady-state solutions of multiple parallel channels in series with impatient customers are obtained by Singh & Ahuja (1995). The solutions of serial and non-serial queuing processes with reneging and balking phenomenon have been studied by Vikram& Singh (1998). The steady-state solution of serial and non-serial queuing processes with reneging and balking due to long queue and some urgent message and feedback phenomenon have been obtained by Singh, Punam & Ashok (2009). In our present society, the impatient customers

generate the most appropriate and modern models in the queuing theory. Incorporating this concept, we study the steady-state analysis of general queuing system in the sense that:

M service channels in series with balking and reneging are linked with N non-serial channels having reneging and balking phenomenon where each of M service channels has identical multiple parallel channels.

The input process is Poisson and the service time distribution is exponential. The service discipline follows SIRO-rule (Service in random order) instead of FIFO-rule

(first in first out). The customer becomes impatient in queue after sometime and may leave the system

without getting service . The input process depends upon the queue size in serial and non-serial channels. Waiting space is finite.

The expression for mean queue length has been obtained whenever the queue discipline is first come first served. The practical situations where such a model finds application are of common occurrence.

2. Formulation of Model:

The system consists of Q i ( i =1,2,….M) service phases where each service phase Q i has ci ( i=1,2,….M) identical parallel service facilities and Q1j channels (j=1,2,….N) with respective servers S i (i=1,2,……M) and S1j (j=1,2,……N).Customers demanding different types of service arrive from outside the system in Poisson distribution with parameters λ i( i =1,2,….M) at Qi service phase and λ1j(j=1,2,…..N) at Q1j service phase respectively . But the sight of long queue at Q i or Q1j , may discourage the fresh customers from joining it and may decide not to enter the service channel Q i or Q1j (j=1,2,….N) then the Poisson input

rates λ i( i =1,2,….M) and λ 1j (j=1,2,….N )would be and where ni and mj are the queue sizes of Qi and Q1j . Further, the impatient customers joining any service channel may leave the queue without getting service after a wait of certain time. The service time distributions for the servers Si

(i=1,2……,M) and S1j (j=1,2…..,N) are mutually independent negative exponential distributions with

service rates and respectively. After the completion of service at Qi (i=1,2,……..M), the customer either leaves the system with probability p i or joins the next

phase with probability such that ; (i= 1,2…..M-1).

After completion of service at QM , the customer either leaves the system with probability or joins

any of the Q1j (j=1,2,…….,N) with probability

qMjm j+1 (j=1,2,….,N) such that =1.

If the customers are more than c i in the Qi service phase ,all the ci servers will remain busy and each

is putting out the service at mean rate and thus the mean service rate at Q i is ci ,on the other hand if the number of customers is less than c i in the Qi service phase ,only ni out of the ci servers will be busy

and thus the mean service rate at Q i is ni (i=1,2,…….M).It is assumed that the service commences instantaneously when the customer arrives at an empty service channel.

3. Formulation of Equations:

Define P(n1 ,n2 , ……….nM ; m1 ,m2,m3, …………mN ;t) as the probability that at time ‘t’ , there are n i customers ( which may balk , renege or after being serviced by the Q i phase either leave the system or join the next service phase ) waiting in the Qi service phase (i=1,2,……..,M) , mj customers ( which may balk or renege or after being serviced leave the system ) waiting before the servers S 1j (j=1,2,…….N) .

We define the operators T i , T i and T i , i+1 to act upon the vector and

T j and T . j and T j , j+1 to act upon the vector as follows:

T i ( ) = (n1,n2,………ni-1…nM)

T i ( ) = (n1,n2,………ni+1…nM)

T i , i+1 ( ) = (n1, n2 ,….,ni +1, ni+1 -1,….,nM )

T j ( ) = (m1,m2,………mj-1…mN)

T . j ( ) = (m1,m2,………mj+1…mN)

T . j , j+1 ( ) = (m1 ,m2 ,……,mj +1,mj+1. -1….mN )

Following the procedure given by Kelly (1979), we write difference differential equations as:

Here we assume that if at any instant, there are K customers in the system , then the customers arriving at that instant will not be allowed to join the system and it is considered as a forced balking and lost for the system.

--------------------------------------------------------(1)

for , and < K; (i = 1,2,3,…,M; j=1,2,3,…,N)

and

-------(2)

for ; and ; (i= 1, 2, 3, …, M; j=1, 2, 3, …, N)

Where

if any of the arguments is negative.

Where are the average rates at which the customers renege after a wait of certain time

whenever there are ni and mj customers in the Qi and Q1j service phases respectively.

are the arrival rates and service rates respectively and if any of the arguments is negative.

4. Steady-State Equations:

We write the following Steady-State equations of the queuing model by equating the time derivatives to zero in the equations (1) and (2)

--------------------------------------------------------------------------------(3)

for ; and

and

-----------(4)

for ; and

Two cases arise depending upon the number of customers and number of channels at phase

(i= 1, 2, 3, …, M)

5. CASE(1)

When the number of customers ni before phase is less than the number of identical service channels

(i.e. < ;i = 1, 2, 3,…, M) , then there is no reneging in and the service is immediately available

to the customers on arrival. Then under such situation = 0 and

Steady State Equations:

The equations (3) and (4) reduce to

--------------------------------------------------------------------------------------- (5)

for ; and

and

-----------(6)

for ; and

Steady State Solutions:

The Steady State solutions of the above equations (5) and (6) can be verified to be

-----------------------------------------------(7)

Where and are as follows:

6. CASE (II)

When the number of customers before phase is more than or equal to the number of identical

service channels (i.e. ), under such situation and and

.

Steady State Equations:

The equations (3) and (4) reduce to

;

--------------------------------------------------------------------------------------(8)

For ni ci ; and

and

----------------------------------------------------------------(9)

for ni ci ; and

Steady State Solution:

The Steady state solution of the above equations can be verified to be:

Where and are as follows:

Here , it is mentioned that the customers leave the system at constant rate as long as there is a line

provided that the customers are served in the order in which they arrive. Putting (j=1,2,3,…,N)

in equations (5), (6),(8) and (9) and in equations (8) and (9), the steady-state solutions (7) and (10) reduce to

………………………………..(11)

for ;

, m j ≥ 0 ; (i=1,2,3,……,M) ; j=1,2,3,……..N).

Where and are the same as mentioned before.

---------------------------------------(12)

for , m j ≥ 0 ; (i=1,2,3,……,M ; j=1,2,3,……..N) .

Where and are the same as mentioned before.

We obtain from the normalizing condition and and with the restrictions that the traffic intensity of each service channel of the system is less than unity.

It has been verified that the results obtained in case of finite waiting space are confirmative with the results of infinite waiting space.

References

1. Hunt, G.C. (1955): Sequential arrays of waiting lines . Opps. Res. Vol. 4, pp674-683.

2. Jackson, R.R.P. (1954): Queuing systems with phase type service. Operations Research Quarterly.

Vol. 5, No.2, pp109-120.

3. Kelly, F.P. (1979): Reversibility and stochastic networks. John Wiley and Sons Inc. New York.

4. Maggu, P.L.(1970): On certain types of queues in series. Statistica Neerlandica. Vol. 24, pp89-975. O’Brien, .G. (1954): The solution of some queuing problems. J.Soc. Ind. Appl. Math. Vol. 2, pp132-

142.

6. Singh, Man.(1984); Steady-state behavior of serial queuing processes with impatient customers.

Math. Operations forsch, U. Statist. Ser. Statist. , Vol. 15. No.2,pp 289-298.

7. Singh, Man and Ahuja,Asha (1996): The Steady State Solutions of Multiple Parallel Channels in

Series With Impatient Customers. Intnl. J. Mgmt. Sys. Vol. 12, No. 1 . pp67-76.

8. Singh Man , Punam and Ashok Kumar (2008): Steady-state solutions of serial and non-serial

queuing processes with reneging and balking due to long queue and some urgent message and

feedback .International Journal of Essential Science Vol. 2, No.2. pp12-17.

9. Vikram and Singh Man (1998): Steady state solution of serial and non – serial queuing processes

with reneging and balking phenomenon. Recent Advances in Information Theory , Statistics and

Computer Applications ,CCS Haryana Agricultural University Publication , Hisar. pp227-236.