Heterogeneous Server Markovian Queue with Partial Breakdown and with Customer Discouragement

Heterogeneous Server Markovian Queue with Partial Breakdown and with Customer Discouragement

Kalyanaraman. 𝐑

and Senthilkumar. 𝐑

1

Department of Mathematics,

Annamalai University, Annamalai Nagar, INDIA.

2

Mathematics Wing, Directorate of Distance Education, Annamalai University, INDIA.

email: [email protected], [email protected]

(Received on: August 27, 2018) ABSTRACT

In this paper we consider a two heterogeneous server Markovian queue with partial breakdown and customer discouragement. If both the servers are ideal, the arriving customer select the fastest server. During busy period the system may breakdown, immediately repair has been carried out. But during the breakdown period the server work in a slower rate. The customers present in the system at the time of a break down may become discouraged and leave the system with constant probability.

This model has been solved using Matrix geometric method. Some performance measures and numerical results are obtained.

2000 Mathematics Subject Classification: 90B22, 60K25 and 60K30.

Keywords: Markovian queue, Heterogeneous server, Steady state solution, Matrix- Geometric method.

1. INTRODUCTION

Queue with breakdown has applications in manufacturing system, production system, telecommunication network and computer system. Single sever queue with server breakdowns have been studied by many researchers including Federgruen and Green (1986), Li et al.

(1997), Tang (1997), Nakdimon and Yechiali (2003), Wang et al. (2007), Wang et al. (2008),

Choudhary and Tadj (2009), to mention a few. Multi-server queueing systems with server

break down has many real life application. However, due to their analytical complexity, there

have been only a few studies carried out on multi-server queueing system with server

breakdowns. The equilibrium analysis for the general input and exponential service time and with n servers was given in Kendall (1953). A non-constructive existence theorem for the stationary distribution of a general input and general service time was presented in Kiefer and Wolfowitz (1955). Karlin and Mc Gregor (1958) obtained the busy period distribution for the M/M/S queue. Krishnamoorthi (1963) considered a Poisson queue with two heterogeneous severs and with violation of the First-in-First-out principle.

Mitrany and Avi-Itzhak (1968), studied an M/M/N queue with server breakdowns and ample repair capacity. In their study, the moment generating function of the queue size is obtained by using the transformation method. Vinod (1985), considered the same model using the matrix-geometric solution method. For N=1, the author imposed some restrictions on the server down-periods (either independent of the queue length or only occurring when the sever is active). Neuts and Lucantoni (1979) and Wartenhosrt (1995) studied the repair model by considering limited repair capacity. Neuts and Lucantoni (1979), considered a single queue of customers, each served by one of N parallel servers. Wartenhosrt (1995) considered N single- server queueing stations, each serving its own stream of customers. Wang and Chang (2002) studied an M/M/R/N queue with balking, reneging and server breakdowns from the viewpoint of queueing. They solved the steady-state probability equations iteratively and derived the steady-state probabilities in matrix form.

In 1999, Abou-El-ata and Shawky introduced a simpler approach to find the condition when to discard the slower server in a heterogeneous two channels queue. Kumar and Madheswari (2005), studied an M/M/2 queueing system with heterogeneous servers and multiple vacations by using the matrix-geometric solution method. They studied the stationary queue length distribution and waiting time distribution along with their means via the rate matrix. Yue et al. (2009), further considered the model in 2005. They obtained the explicit expression of the rate matrix and proved the conditional stochastic decomposition results for the stationary queue length and waiting time. Madan et al. (2003), studied a two-server queue with Bernoulli schedules and a single vacation policy where the two servers provide heterogeneous exponential services to the customers. They obtained steady-state probability generating functions of the system size for various states of the servers.

In real life queueing situations it can be seen that the customers leave the system without getting service after a prolonged wait. In the queueing literature this customer behavior is called customer impatience. Also there are situations, in which if the system is not operational, the customer leaves the system without getting service, this is called customer discouragement. This type have been thoroughly reviewed in Doshi (1986). The reader may refer Daley (1965), Kok and Tijms (1985) and Chan et al. (1993).

In this paper we consider a two heterogeneous server Markovian queue with partial

breakdown. If both the servers are ideal, the arriving customer selects the fastest server. During

busy period the system may breakdown, immediately repair has been carried out. But, during

the breakdown period each customer present in the system may become discouraged and leave

with constant probability independently of other customers. Also, during the breakdown

period the server work on a slower rate. The model has been defined in section 2 and analyzed

in section 3. Some performance measures are given in section 4, a numerical study has been carried out in section 5. A conclusion has been given in the last section.

2. THE MODEL

We consider an M/M/2 queueing model with heterogeneous servers, server 1 and server 2. The inter arrival time of customers follows negative exponential distribution with mean

. Server 1 and 2, served the customers based on exponential distribution with rates µ

and µ

respectively (µ

< µ

) and let µ = µ

+ µ

. Each customer is served only by one server and the queue discipline is first come first served. If the system is empty arriving customer always joins the first server. During service, the system may breakdown, the breakdown period follows negative exponential distribution with rate α. Immediately the repairing process starts, the repair period follows negative exponential distribution with rate β. During the breakdown period, the servers serve the customers with lower rate µ

(server 1) and µ

(server 2) and let µ

= µ

+ µ

(µ

< µ

< µ

< µ

). During break down/repair periods customers still arrive at Poisson rate λ, but may become discouraged and fail to join the queue with probability (1 − 𝜃), independently of the state of the system. If an arriving customer finds both the servers busy the customer waits in a waiting line of a infinite capacity for the first free server.

Let𝑋(𝑡) = (𝐿(𝑡), 𝐽(𝑡)), then {(𝑋(𝑡)): 𝑡 ≥ 0} is a Continuous time Markov chain (CTMC) with state space 𝑆 = {(𝑖, 𝑗): 𝑖 = 0,1; 𝑗 ≥ 0}, where j denotes the number of customer in the system and i denotes the server state and (0, 0) = 0.

Using the lexicographical sequence for the states, the rate matrix Q, is the infinitesimal generator of the Markov chain and is given by

𝑄 =

[

C

𝐶

𝐵

B

𝐴

𝐴

𝐴

𝐴

𝐴

𝐴

𝐴

. . . . . . . . .]

where the sub-matrices 𝐴

, 𝐴

, and 𝐴

are of order 2 x 2 and are appearing as 𝐴

= [𝜆𝜃 0 0 𝜆 ]

𝐴

= [ −(𝜆𝜃 + µ

+ 𝛽) 0

0 −(𝜆 + µ) ]

𝐴

= [ µ

0 0 µ]

and the boundary matrix is defined by

𝐶

= −(𝜆)

𝐶

= (0 𝜆) 𝐵

= [ 0

µ

]

𝐵

= [ −(𝜆𝜃 + 𝛽) 𝛽

𝛼 −(𝜆 + 𝛼 + µ

)]

3. THE ANALYSIS

The model defined in this section 2 can be studied as a quasi-birth-and-death (QBD) process. For the analysis the following notation have been introduced. At time t, let N(t) be the number of customers in the system and J(t) the server state.

𝐽(𝑡) = { 0, 𝑖𝑓 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑏𝑟𝑒𝑎𝑘𝑑𝑜𝑤𝑛 1, 𝑖𝑓 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑏𝑢𝑠𝑦 𝑠𝑡𝑎𝑡𝑒

Let 𝑃 = (𝑝

, 𝑝

, 𝑝

, … ) be the stationary probability vector associated with Q, such that PQ=0 and Pe=1, where e is a column vector of 1's of appropriate dimension,

where 𝑝

= (𝑝

), 𝑝

= (𝑝

, 𝑝

) for 𝑖 ≥ 1.

If the steady state condition is satisfied, the sub vectors 𝑝

satisfies following equations:

𝑝

𝐶

+ 𝑝

𝐵

= 0 (1)

𝑝

𝐶

+ 𝑝

𝐵

+ 𝑝

𝐴

= 0, (2)

𝑝

𝐴

+ 𝑝

𝐴

+ 𝑝

𝐴

= 0, 𝑖 ≥1 (3)

𝑝

= 𝑝

𝑅

; 𝑖 ≥ 2 (4)

where R is the rate matrix, is the minimal non-negative solution of the matrix quadratic equation (see Neuts(1981)). 𝑅

𝐴

+ 𝑅𝐴

+ 𝐴

= 0 (5)

Substituting the equation (4) in (2), we have 𝑝

𝐶

+ 𝑝

(𝐵

+ 𝑅𝐴

) = 0 (6)

and the normalizing condition is 𝑝

+ 𝑝

(𝐼 − 𝑅)

𝑒 = 1 (7) Theorem 1. The queueing system described in the article is stable if and only if 𝝆<1, where 𝜌 =

Proof. Consider the infinitesimal generator𝐴 = [−𝛽 𝛽

𝛼 −𝛼 ], which is a square matrix of order 2, the row vector 𝜋 = (𝜋

, 𝜋

)satisfying the condition 𝝅A=0 and 𝝅e=1.

That is,

The system is stable if and only if

< 1

Theorem 2. If 𝝆<1, the matrix equation (5) has the minimal non-negative solution

𝑅 = −𝐴

𝐴

− 𝑅

𝐴

𝐴

Proof. We define the matrix 𝐴 = 𝐴

+ 𝐴

+ 𝐴

. This matrix A is a 2 x 2 matrix and it can be written as A= [−𝛽 𝛽

𝛼 −𝛼 ]

A is reducible. The analysis present in Neuts (1978) is not applicable. In Lucantoni (1979), similar reducible matrix is treated for the case when the elements are probabilities.

Equation (5), can be written as, 𝐴

𝐴

+ 𝑅𝐴

𝐴

+ 𝑅

𝐴

𝐴

= 0𝐴

Since 𝐴

is non-singular, 𝐴

exists and

𝑅 = −𝐴

𝐴

− 𝑅

𝐴

𝐴

(8) where

𝐴

= 1

[(𝜆 + 𝛼 + µ)(𝜆𝜃 + 𝛽 + µ

) − 𝛼𝛽] [ −(𝜆 + 𝛼 + µ) −𝛽

−𝛼 −(𝜆𝜃 + 𝛽 + µ

) ]

Using Neuts and Lucantoni (1979), the matrix R is numerically computed by using the recurrence relation with R (0) =0 in equation (8).

Theorem 3. If 𝝆<1, the stationary probability vectors 𝑝

and 𝑝

= (𝑝

, 𝑝

) are

𝑝

= 1

[1 + (

0)(1−𝑟1)−𝑟10𝑟₀₁

) (

1(𝜆𝜃+𝛽+µ^′𝑟₀)

+

1

)]

𝑝

= [ 𝜆(µ

𝑟

+ 𝛼) µ

(𝜆𝜃 + 𝛽 − µ

𝑟

) ] 𝑝

𝑝

= ( 𝜆

µ

) 𝑝

and 𝑝

= 𝑝

𝑅

; 𝑖 ≥ 2

Proof. 𝑝

, 𝑝

and 𝑝

follows from the equations (1), (2) and (7).

Remark 1. Even though R in Theorem 2 has a nice structure which enables us to make use of the properties like𝑅

= [ 𝑟

𝑟

∑

𝑟

𝑟

0 𝑟

], for 𝑛 ≥ 1due to the form of 𝑟

& 𝑟

, it may not be easy to cary out the computation required to calculate the 𝑝

and the performance measures. Hence, we explore the possibility of algorithmic computation of R. The computation of R can be carried out using a number of well-known methods. We use Theorem 1 of Latouche and Neuts (1980). The matrix R is computed by successive substitutions in the recurrence relation:

𝑅(0) = 0 (9)

𝑅(𝑛 + 1) = −𝐴

𝐴

− [𝑅(𝑛)]

𝐴

𝐴

for 𝑛 ≥ 0 (10) and is the limit of the monotonically increasing sequence of matrices {𝑅(𝑛), 𝑛 ≥ 0}.

4. PERFORMANCE MEASURES

Using straightforward calculations the following performance measures have been

obtained:

(i) probability of down time = 𝑝

(ii) probability that both the servers be busy = 1 − 𝑝

− (𝑝

+ 𝑝

) (iii) Expected number of customers in the system 𝐸(𝑁) = ∑

𝑛𝑝

𝑒

(iv) Expected number of customer served 𝐸(𝑀) = µ

𝑝

𝑒 + (µ

+ µ

) ∑

𝑝

𝑒 (v) Expected waiting time of a customer in the system, according to Little's law, is 𝐸(𝑊) = 𝐸(𝑁)

𝜆 5. NUMERICAL STUDY

In this section, some examples are given to show the effect of the parameters λ, µ

, µ

, µ

, µ

, α, β and θ on the probability of down time, probability that both the servers be busy, Expected number of customers in the system, Expected number of customer served, Expected waiting time of a customer in the system for the model analyzed in this paper. The corresponding results are presented as case (1), case (2).

Case (1): If λ=0.3, µ

= 4.5, µ

= 3.2, µ

= 2.5, µ

= 1.2, α=0.2, β=0.5 and θ=0.8 the matrix R is obtained using the equations (9) & (10)

𝑅 = [0.056933 0.003812 0.001862 0.038066 ]

and the invariant probability vector is 𝑃 = (𝑝

, 𝑝

, 𝑝

, … ) where

𝑝

= (0.904346)

𝑝

= (0.023840, 0.060996)

and the remaining vectors 𝑝

𝑠 are evaluated using the relation 𝑝

= 𝑝

𝑅

; 𝑖 ≥ 2 𝑝

= (0.001470857300, 0.002412751783)

𝑝

= (0.000088232868, 0.000097450720) 𝑝

= (0.000005204815, 0.000004045903) 𝑝

= (0.000000303859, 0.000000173852) 𝑝

= (0.000000017623, 0.000000007776) 𝑝

= (0.000000001018, 0.000000000363) 𝑝

= (0.000000000059, 0.000000000018) 𝑝

= (0.000000000003, 0.000000000001)

For the chosen parameters 𝑝

→ 0, and the sum of the steady state probabilities is found to be 0.993862

The performance measures are

(i) probability of down time 𝑝

= 0.904346

(ii) probability that both the servers be busy = 0.010818

(iii) Expected number of customers in the system E(N)= 0.093197 (iv) Expected number of customer served E(M)= 0.4131703

(v) Expected waiting time of a customer in the system, according to Little's law, is

E(W)= 0.310657

Case (2): If λ=0.5, µ

= 5.2, µ

= 4.6, µ

= 3.3, µ

= 2.2, α=0.3, β=0.6 and θ=0.8 the matrix R is obtained using the equations (9) & (10)

𝑅 = [0.065352 0.004139 0.002536 0.049597 ]

and the invariant probability vector is 𝑃 = (𝑝

, 𝑝

, 𝑝

, … ) where

𝑝

= (0.864316)

𝑝

= (0.041203, 0.084069)

and the remaining vectors 𝑝

𝑠 are evaluated using the relation 𝑝

= 𝑝

𝑅

; 𝑖 ≥ 2 𝑝

= (0.002905897563, 0.004340109415)

𝑝

= (0.000200912735, 0.000227283905) 𝑝

= (0.000013706442, 0.000012104178) 𝑝

= (0.000000926440, 0.000000657062) 𝑝

= (0.000000062211, 0.000000036423) 𝑝

= (0.000000004158, 0.000000002064) 𝑝

= (0.000000000277, 0.000000000120) 𝑝

= (0.000000000018, 0.000000000007)

For the chosen parameters 𝑝

→ 0, and the sum of the steady state probabilities is found to be 0.996289

The performance measures are

(i) probability that down time 𝑝

= 0.864316

(ii) probability that both the servers be busy = 0.010412

(iii) Expected number of customers in the system E(N)= 0.141156 (iv) Expected number of customer served E(M)= 0.726884

(v) Expected waiting time of a customer in the system, according to Little's law, is E(W)= 0.282312

6. CONCLUSION

In this article, a heterogeneous two-server queueing system with partial breakdown and with customer discouragement has been studied. The model studied in this article is more realistic for modeling queueing situations where the server may experienced many types of breakdowns which can be realized in manufacturing production systems. The system can be generalized by taking 𝐶 ≥ 3 customers.

REFERENCES

1. Abou-El-Ata, M.O., and Shawky, A.L., A simple approach for the slower server problem, Commun Fac. Sci, uni. Anv series A1; 88: 1-6 (1999).

2. Chan, W., Bartosynski, R., and Pearl, D., Department of Statistical Technical Report, No.414, The Ohio State University, (1989).

3. Choudhury, G., and Tadj, L., An M/G/1 queue with two phases of service subject to the

server breakdown and delayed repair, Applied Mathematical Modelling; 33: 2699-2709 (2009).

4. Daley, D.J., General customer impatience in the queue GI/G/1, J.Appl.Probab.; 2: 186- 205 (1965).

5. Doshi, B.T., Queueing systems with vacations- a survey, Queueing Systems; 1: 29-66 (1986).

6. Federgruen, A., and Green, L., Queueing systems with service interruptions, Operations Research; 34: 752-768 (1986).

7. Karlin, S., and Gregor, J.Mc., Many server queuing process with Poisson input and exponential service times, Pacific J.Math.; 8: 87-118 (1958).

8. Kendall, D.G., Stochastic process in the theory of queues, Ann. math. stat.; 24:333-354 (1953).

9. Kiefer, J., and Wolfowitz, J., On the theory of queues with many servers, Trans Amer.

math. soc.; 781-18 (1955).

10. Kok, A.G.De., and Tijms, H.C., A queueing system with impatient customers, J. Appl.

Probab.; 22: 688-696 (1985).

11. Krishnamoorthi, B., On Poisson queue with two heterogeneous servers, Oper. Res.; 11(3):

321-330 (1963).

12. Kumar, B.K., and Madheswari, S.P., An M/M/2 queueing system with heterogeneous servers and multiple vacations, Mathematical and Computer Modelling; 41: 1415-1429 (2005).

13. Li, W., Shi, D., and Chao, X., Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations, Journal of Applied Probability; 34: 546-555 (1997).

14. Lucantoni, D.M., A 𝐺𝐼/𝑀/𝐶 queue with a different service rate for customers who need not wait an algorithmic solution, Technical Rep. Univ. of Delware,USA (1979).

15. Madan, K.C., Abu-Dayyeh, W., and Taiyyan, F., A two server queue with Bernoulli schedules and a single vacation policy, Applied Mathematics and Computation; 145: 59- 71 (2003).

16. Mitrany, I.L., and Avi-Itzhak, B., A many-server queue with service interruptions, Operations Research; 16: 628-638 (1968).

17. Nakdimon, O., and Yechiali, U., Polling systems with breakdowns and repairs, European Journal of Operational Research; 149: 588-613 (2003).

18. Neuts, M.F., Markov chains with applications in queueing theory which have a matrix- geometric invariant probability vector, Adv. Appl. Probab; 10: 185-212 (1978).

19. Neuts, M.F, Matrix-Geometric solution in stochastic models, Vol 2 of John Hopkins series in the Mathematical Sciences, Johns Hopkins University press, Baltimore, md, USA, 1981.

20. Neuts, M.F., and Lucantoni, D.M., A Markovian queue with N servers subject to breakdowns and repairs, Management Science; 25: 849-861 (1979).

21. Tang, Y., Single-server M/G/1 queueing system subject to breakdowns-some reliability and queueing problems, Microelectronics Reliability; 37: 315-321 (1997).

22. Vinod, B., Unreliable queueing systems, Computers and Operations Research; 12: 322- 340 (1985).

23. Wang, K.H., and Chang, Y.C., Cost analysis of a finite M/M/R queueing system with balking, reneging and server breakdowns, Mathematical Methods of Operations Research;

56: 169-180 (2002).

24. Wang, J., Liu, B., and Li, J., Transient analysis of an M/G/1 retrial queue subject to disasters and server failures, European Journal of Operational Research; 189: 1118-1132 (2008).

25. Wang, K., Wang, T., and Pearn, W., Optimal control of the N policy M/G/1 Queueing systems with server breakdowns and general startup times, Applied Mathematical Modelling; 31: 2199-2212 (2007).

26. Wartenhosrt, P., N parallel Queueing systems with server breakdowns and repair, European Journal of Operational Research; 82: 302-322 (1995).

27. Yue, D., Yu, J., and Yue, W., A Markovian queue with two heterogeneous server and

multiple vacations, Journal of Industrial and Management Optimization; 5: 453-465

(2009).