Efficient Graduate Employment Serving System based on Queuing Theory

Transcription

1 76 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER Efficient Graduate Employment Serving System based on Queuing heory Zeng Hui School of Sciences, Yanshan University, Qinhuangdao, China Abstract he mathematical model of an two-phases-service M/M// queuing system with the server breakdown and multiple vacations was realized and established in the Graduate Employment Services system. Secondly, equations of steady-state probability were derived by applying the Markov process theory. hen, we obtained matrix form solution of steady-state probability by using blocked matrix method. Finally, some performance measures of the system such as the expected number of users in the system and the queue were also presented. Index erms queuing theory, mathematical model, Graduate Employment Services system I. IRODUCIO Queuing theory is a branch of operations research. he main purpose of the study is to answer how to improve the service provided to an object making a cerain indicator target to achieve optimal []. Queuing theory originated in 99 from Copenhagen, Dennark elephone Company s A. K Erlang s famous paper robability theory and phone calls []. he thesis of the paper was focused on phone calls creating an applied mathematics in this subject, and many basic principles of the discipline. At present, domestic and international use of queuing theory in the optimization of window and communication facilities. Many researchers have proposed estimation methods of the number of units in circulation and network bandwidth based on queuing theeory.[3, 4]. In recent years, many scholars began to study real-life problems of queuing theory, such as the application in the arrangements in hospital clinics and wards [5, 6], an optimized method for the loading/unloading system of port transportation [7], the application in determining the number of bank teller window and staff [8], the supermarket checkout queue management [9, ], and a variety of after-sales service system []. However, we have not seen any papers about the analysis of Graduate Employment Service system. he models above only study the case of one service per user provided by each server. In fact, in our daily life, we often encounter a server offering different services for the same users. In such queuing models, all the users need the first phase service and only part of them will be asking the server to provide a second phase service, which is the two-phases-service queuing system. Recently, there have been several contributions considering queuing system in which the server may provide a second phase service. Madan [] studied an M/G/ queue with the second optional service in which first essential service time follows a general distribution but second optional service is assumed to bi exponentially distributed. Medhi [3], generalized the model by considering that the second optional service is also governed by a general distribution. Yue Dequan [4-7], studied an M/M// queue with the multiple vacations; they obtained the matrix form solution of steady-state probability. he system considered in this paper is the secondary server system which is mentioned above. Graduate Employment Services system is an information service platform that facilitates student s employment and has a lot of queues in it, and therefore the issue of seeking optimal solution also exists. In this paper we will take a college student employment service system that can accommodate a limited number of users for example, and consider a college student employment service system model in which the server can provide two phases service, and takes a vacation when the system becomes idle. Once service begins, the service mechanism is subject to breakdowns. arameters are calculated based on actual data and validated to study the performance of its services. II. SYSEM MODE Graduate Employment Services system capacity is. Suppose there are only one help desk system for service users. Under normal circumstances, access to the query must be employed to process information, that is, users must first accept the first phase of service. Subsequently, the user will then submit resume online based on their needs, which means choosing to accept the second phase of service. he service mechanism may fail. A. Input process In a certain period of time, each user can repeatedly enter the system, so the source of users can be seen as a infinite populaion. Users arrive independently according to oisson process with different rates. Arrival rate during vacation is λ, arrival rate during active service is λ, arrival rate during breakdown is λ. ACADEMY UBISHER doi:.434/jcp

2 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER 77 B. Queuing discipline he first essential service is needed for all arriving users. he vacation times, uninterrupted service times, and the repair times follow exponential distribution. he first service rate is μ, and the second service rate is μ. As soon as the first service of a customer is completed, then with probability θ ( < θ < ), he may opt for the second service, in which case his second service will immediately commence or else with probability θ, he may opt to leave the system, in which case another customer at the head of the queue is taken up for his first essential service. C. Service rules he server goes on vacation instantly when the queue becomes empty, and continues to take vacations of exponential length until, at the end of a vacation, users are found in the queue. he vacation rate is v and vacation time follows exponential distribution. Service mechanism breakdowns occur only during the first active service, and the breakdown rate is b ( < b < ).he service mechanism goes through a repair process of random duration, and once repair is completed, the server returns to the customer whose service was interrupted, the repair rate is r.various stochastic processes involved in the system are assumed independent of each other. III. SEADY-SAE ROBABIIY EQUAIOS et X () t be the number of customers in the system at time t. Define Ct () as the state of the server at the time t. And define the state as follows: ( t) ( t) ( t) ( e t), he server is on vacation at the time, he server is on the first service at the time Ct () =, he server is on the second service at the time 3, he server is on breakdown process at the tim hen, { X( t), C( t), t } is a Makov process with state space as follows: Ω= {( n,) : n } U {( n, j) : n, j =,,3} he steady-state probability of the system is defined as follows: p ( n) = lim p( X( t) = n, C( t) = ), n t p ( n) = lim p( X( t) = n, C( t) = j), n j t By applying the Makov process theory, we can obtain the following set of steady-state probability equations: λ p () = μ ( θ) p () + μ p (), () ( v+ λ ) p ( n) = λ p ( n ), n () ( μ + λ + bp ) () ( μ + λ + bp ) ( n) = vp () + μ ( θ) p () + μ p () + rp (), (4) 3 = vp ( n) + λ p ( n ) + μ ( θ) p ( n + ) 3 + μ p ( n+ ) + rp ( n), n (5) ( μ b) p vp λ p rp + = + ( ) +, (6) 3 ( μ + λ ) p () = μθ p (), (7) ( μ + λ ) p ( n) = μθ p ( n) + λ p ( n ), n (8) μ p = μθp + λ p ( ), (9) ( r+ λ ) p () = bp (), () ( r+ λ ) p ( n) = bp ( n) + λ p ( n ), n () rp = bp + λ p ( ), () 3 3 p ( n) + p ( n) + p ( n) + p ( n) =. (3) 3 n= n= n= n= IV. MARIX FORM SOUIO In the following, we derive the steady-state probability by using the partitioned block matrix method. et = ( p(),,,, 3) be the steady-state probability vector of the transition rate matrix Q, where ( (), (),, ) = p p p ( (), (),, )(, 3) = p p p i i i i i hen, the steady-state probability equations above can be rewritten in the matrix form as Q = e = (4) Where e is a column vector with 4 + components, and each component of e equal to one, and the transition rate matrix Q of the Markov process has the following blocked matrix structure: vp = λ p ( ), (3) ACADEMY UBISHER

3 78 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER λ η A B Q = α B C D β B C B3 D 3 Where e is a column vector with 4 + components, and each component of e equal to one, and the transition rate matrix Q of the Markov process has the following blocked matrix structure: λ η A B Q = α B C D β B C B3 D 3 Each sub-matrix of the matrix Q as follows: v + λ λ v + λ λ A = M M M M M v + λ λ v μ+ λ+ b λ μ( θ) μ+ λ+ b μ ( θ) B = M M M M λ μ+ λ + b λ μ( θ) μ+ b C μ B = μ M M M M μ μ + λ λ μ + λ M M M M = λ μ + λ λ μ r + λ λ r + λ M M M M D3 = λ r + λ λ r B = vdiag,, K, B3 = rdiag,, K, C = μθ diag,, K, D = bdiag,, K, Where λ is a constant, η = ( λ,,,) is a row vector, ( ),,, α = μ θ, β = ( μ,,,) are column vectors. A, B ( ), ( ), ( 3 i i Ci i Di i ) are square matrices.eq. (4) is rewritten as follows: λ p + α + β =, (5) p η + A =, (6) B + B + B + B 3 3 =, (7) C C, + = (8) D + = (9) D 3 3, p + e + e + e + e =, () 3 Where e is column vector with components, and component of e to one. From Eq. () we get λ p () = p () () + v λ λ p( k) = p() v + λ k ( k ) () λ λ p = p() v v+ λ (3) ACADEMY UBISHER

4 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER 79 From Eq. (8), we get From Eq. (9), we get = CC (4) = DD (5) 3 3 Substituting Eq. (4) and (5) into (7), we get v B CC B DD B = v 3 3 λ, λ,, λ λ, λ = v+ λ v+ λ v+ λ v v+ λ p (6) et A = B CC B DD 3 B3, after some algebraic manipulation we find the component of the A as follows: λ br μ λ μ μ θ μ λ λ μ, i = j = μ j i j + + b +, i = j ( + ) r + μ ( θ), = +, μ + λ μ ( θ), i =, j = j + i aij = μλ brλ λ, i j j + i + < ( μ + λ) ( r + λ) brλ, i =, j = ( r + λ ) brλ λ +, i =, j = ( r + λ ), others r et A = A%, each sub-matrix of the matrix A r as follows: a a a3 a4 a5 a6 a( ) a3 a33 a34 a35 a36 a3 ( ) a43 a44 a45 a46 a4( ) A M M O O O O O M % = a( 3)( 4) a ( 3)( 3) a( 3)( ) a ( 3)( ) a( )( 3) a( )( ) a ( )( ) a( )( ) a ( )( ) a ( ) r = ( a, a, a3,, K,) is a ( ) =, K,,, is a ( ) ( ( ) ) r a a vector. et Where ( (), ) = p %, ( (, ) ( 3, ), ) % = p p K p. row vector, column Eq. (4) is rewritten as follows: p() r+ A % % = v( p(), p( ), K, p( ) ) = vσ p (7) where λ = = v + λ λ r % vp p (8) λ λ λ σ =,,, v+ λ v+ λ v+ λ heorem. A % is an invertible matrix, the determinant is aii ( ) i= A% = roof. Obviously, A % is an upper triangular matrix, the determinant is equal to the product of diagonal elements, that is A% = aii ( ). For i= λ, λ, λ, μ, μ, vbr,, >, < b <, λ, λ, According to the above expression of the a ij, that a <, i =,3, K,, so A%. ii From theorem and Eq. (6), we get % v p A = σ % p ra% (9) Substituting Eq. (9) into (8), ACADEMY UBISHER

5 8 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER we get where λ p = v A r p % () σ % λ ra v r λ + = cp (3) λ c = vσa% r λ ra% r v + λ Substituting Eq. (3) into (9), we get ( σ % % ) is a constant. % = p v A cra (3) et q = vσ A % cra % is a that % = p q. So ( (), ) ( )(, ) row vector, = p % = p c q (3) Substituting Eq. (3) into (4), we get ( )(, ) = μθp c q C (33) Substituting Eq. (3) into (5), we get ( )(, ) = bp c q D (34) 3 3 Substituting Eq. (6), (33) and (34) into (), we get So p = + η p p A e ( )(, ) μθ ( )(, ) + p cq e + p cq C e +, = bp c q D3 e + ηa e + c, q e + c, q C e + b c, q D e μθ 3 (35) Substituting Eq. (35) into (6), (33) and (34), we get the matrix solution of,, 3. In summary, we have the following theorem. heorem. robability matrix of the steady state solution is: p = δ Where ( c q) = δ, = μθδ ( c, q) C (, ) = bδ c q D 3 3 δ = m μθ m= + ηa e + c, q e + c, q C e + b c, q D e 3 V. ERFORMACE MEASURES OF SYSEM A. he robability hat the System Service Station During Busy eriod B n= n= n= δ (, ) μθ(, ) = p n + p n = c q + c q C B. he robability hat the System Service Station During Vacation eriod V = p ( n) = A δ η εn+ n= n= C. he Average Waiting Queue ength of the System ( q ) = + ( + ) E np n np n n= n= + np n + + np n + 3 n= n= ( cq, ) n + (, ) + bcqd (, ) ε μθ cqc ε + n+ = δ n + δ n= 3 εn+ ηa εn+ D. he Average Queue ength of the System = + E np n np n n= n= np( n) np3( n) + + n= n= = δ nηa ε + δ n= n+ = δη A n= (, ) (, ) (, ) + δ n c q ε n + μθ c q C εn + b c q D3 ε n ACADEMY UBISHER

6 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER 8 et ε n be a column identity vector of order with is n th component equals to one and the other components equal to zero. VI. GRADUAE EMOYME SERVICES SYSEM M/M// QUEUIG MODE OF HE CASE SUDY Based on the above analysis, we obtain the average waiting queue length and the average queue length of the graduate employment services system, and some other state indicators. But as a management decision makers not only to know the steady-state targets, but also to understand some of the parameters on the impact of these state indicators of the system, so that the queuing system as optimal. We take = 5 for example, when λ = λ = λ =, μ =, b =.5, v =, r =, μ and θ Impact on the average queue length of the system. increase will find that attendant faster and faster, steady state system in reducing the number of customers. When 3 λ λ <, E( q ) changes faster. hen, with the increase, the increase of the E( q ) gradually slows down. In Figure, we fix μ = μ =, λ = λ =, v =, b =, r =, θ =.5. Consider when user s arrival rate λ changes, the average queue length of changes. ooking at Figure, with the λ increase will find that attendant faster and faster, steady state system in increasing the number of customers. When λ < 3, E( ) changes faster. hen, with the λ increase, the increase of the E( q ) gradually slows down. Figure. he expected waiting queue length E( q ) vs. the arrival rate λ Figure 3. he expected waiting queue length E( q ) vs. the first busy service rate μ Figure. he expected queue length E() vs. the arrival rate λ In Figure, we fix μ = μ =, λ = λ =, v =, b =, r =, θ =.5. Consider when user s arrival rate λ changes, the expected waiting queue length of changes. ooking at Figure, with the λ Figure 4. he expected queue length E() vs. the first busy service rate μ In Figure 3, we fix λ = λ = λ =, μ =, v =, b =, r =, θ =.5. Consider when the first busy service rate μ changes, the expected waiting queue length of ACADEMY UBISHER

7 8 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER changes. ooking at Figure 3, with the μ increase, we will find E( q ) first decreases rapidly, steady state system in reducing the number of customers. When μ > 6, the increase of the E( q ) gradually slows down. In Figure 4, we fix λ = λ = λ =, μ =, v =, b =, r =, θ =.5. Consider when the first busy service rate μ changes, the average queue length of changes. ooking at Figure 4, with the μ increase, we will find decreases rapidly at first, then in equilibrium. E( ) In Figure 6, we fix λ = λ = λ =, μ = =, μ v =, b =, r =, Consider when the probabilityθ of the users chose the second service changes, the average queue length of changes. ooking at Figure 6, with the number of the user who chose the second service increase, we will find E( ) increases linearly with increasing trend. Figure 7. he expected queue length E() vs. the service rate μ and θ Figure 5. he expected waiting queue length E( q ) vs. the probability θ of the users chose the second service In Figure 5, we fix λ = λ = λ =, μ = =, μ v =, b =, r =, Consider when the probabilityθ of the users chose the second service changes, the expected waiting queue length of changes. ooking at Figure 5, with the number of the user who chose the second service increase, we will find E( q ) increasing increases linearly with trend. In Figure 7, we fix λ = λ = λ =, μ =, b =.5, v =, r =, and μ from.5 to.5, θ from to. ooking at Figure, with the μ increase will find that attendant faster and faster, steady state system in reducing the number of customers. When μ is fixed, with the θ increases, the average queue length gradually increases. Figure 8. he expected queue length E()vs.the service rate μ and b Figure 6. he expected queue length E() vs. the probability θ of the users chose the second service In Figure8, we fix λ = λ = λ =, μ =, θ =.5, v =, r =, and μ from.5 to.5, b from to. ooking at Figure, with the μ increase will find that attendant faster and faster, steady state system in reducing ACADEMY UBISHER

8 JOURA OF COMUERS, VO. 7, O. 9, SEEMBER 83 the number of customers. When μ is fixed, with the b increases, the average queue length gradually increases. hrough the above analysis, we could get a clearer understanding of the system and some of the parameters on the performance of queuing systems. Using this result, service providers can design a reasonable rate and holiday vacation service rate so that the queuing system could achieve as optimal. VII. COCUSIO Queuing model can be used to the employment services system and its design and to optimize the actual system according to the specific requirements of the system. Queuing system is suitable for analyzing and studying random phenomenon such as the employment service system services. In this paper, the Graduate Employment Services system service queuing model can effectively assess the situation, and support the decisionmaking with regards to the management and services of the university employment service system. REFERECES [] Sun Ronghuan, i Jianping, he Basis of Queuing heory, eking: Science,, pp. -7. [] Zhang Rui, Analysis of the queuing theory of service industry, Journal of Qiqihar University hiliosophy, vol. 6, pp. 4-43,. [3] Wolff R W, Stochastic Modeling and the heory of Queues, ew York: rentice Hall,, pp [4] Meng Yuke, Basic and Applied Queuing heory, Shang Hai:ongji University, 989, pp. 7-. [5] Yang Feng, iu Di, Queuing theory to improve patient management in the application queue, University Science Research, vol. 6, pp. 8-9, [6] iu Zhan, Xuyange, he application of queuing theory in the eye s hospital beds, China ew echnologies and roducts, vol. 5, pp ,. [7] Huang Daming, Wen Bing, Jiang Shunmei, An optimized method for the loading/unloading system of port transportation based on queuing theory, Journal of Guangxi University. at Sci Ed, vol. 34, pp , 9. [8] Sun Zhonghui, he application of queuing theory in the bank and the teller window, Operation and Management. vol. 6, pp. -,. [9] Qin i, he application of queuing theory in supermarket checkout service system, Modern Economy, vol., pp. 7-8, 9. [] Gao Yingying, Zhou Jingzhen, Qian ing, he application of queuing theory in library s service marketing, SCI-ECH Information Development & Economy, vol. 4, pp. 3-5,. [] Kong Xiangping, he application of queuing theory in the library circulation services system, he ibrary Journal of Shangdong, vol., pp. 88-9,. [] K.C. Madan, An M/G/ queue with second optional service, Queue. Syst, vol. 34, pp ,. [3] J. Medhi, A single server poisson input queue with a second optional channel, Queue. Syst., vol. 4, pp. 39-4,. [4] Yue D, Zhang Y, Optimal performance analysis of an M/M// queue system with balking, reneging and server vacation, International Journal of ure and Applied Mathematics, vol. 8, pp. -5, 6. [5] ian R, Yue D, Hu, M/ H / Queuing System with Balking, -olicy and Multiple Vacations, Operation Research and Management Science, vol. 4, pp. 56-6, 7 [6] Yue D, Sun Y, he Waiting ime of the M/M// Queuing System with Balking Reneging and Multiple Vacations, Chinese Journal of Engineering Mathematics, vol. 5, pp , 8. [7] Yue D, Sun Y, he waiting time of M/M/C/ queuing system with balking, reneging and multiple synchronous vacations of partial servers. Systems Engineering heory & ractice, vol., pp , 8. Zeng Hui, borrn in January 98 in Wangqing, China. She graduated from Yanshan University of China, and accessed to the Master Degree of science. She is mainly engaged in research in the area of queuing theory. ecturer. ACADEMY UBISHER