關於具有優先順序但無耐心等候之多服務員排隊模型 - 政大學術集成

全文

(1)國立政治大學應用數學系碩士學位論文. 政治大. ‧ 國. 學. On Two Priority Multi-Server Queues 立 with Impatient Customers ‧. 關於具有優先順序但無耐心等候之多服務員排隊模型 n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 碩士班學生：李泓緯撰指導教授：陸行博士中華民國 105 年 5 月 19 日.

(2) Abstract This study considers a system of multi-server queues with two classes of impatient customers: high-priority and low-priority. Customers join the system according to a Poisson process and customers may abandon service after entering the queue for an exponentially distributed duration with distinct rates. In this thesis, we consider last come - first served (LCFS) and first come - first served (FCFS), and service time is assumed to be distributed exponentially among all customers. Deriving the Laplace transforms of the defined random variables and applying the matrix geometric method with direct truncation makes it possible to obtain an approximation of the stationary distribution in order to calculate the expected waiting time for both classes of customers. For each class of customer, we derive performance measures related to stationary probability distributions and conditional waiting times. Keywords: multi-server queues, impatient customers, non-preemptive service policy. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i. i n U. v.

(3) 中文. 要. 我們考慮多服務員的排隊系統，其中包含兩種沒耐心的顧客群，分別是高優先級和低優先級的顧客。顧客的到來滿足布阿松過程，顧客群有一個滿. 政治大務時間服從指數分配。所有的顧客和服務類型分為先來先服務（First Come 立足指數分配的耐心程度，在超出這段時間後，會離開系統。在本文中，服. ‧ 國. 學. First Served, FCFS）和後來先服務（Last Come First Served, LCFS）兩種。藉由隨機變數的拉普拉斯轉換和矩陣幾何方法配合截取法，得到一個近似值的. ‧. 機率分配。我們會計算兩顧客群等候時間的期望值。並且對於較重要的高優. y. Nat. 先級顧客，計算他們有給定條件的期望值。. n. al. er. io. sit. 關鍵字：多服務員等候系統、無耐心排隊、非搶占優先服務策略. Ch. engchi. ii. i n U. v.

(4) Contents. Abstract. 立. Contents. ‧ 國. Preliminaries. sit. 2. io. Introduction. v vi. y. Nat. 1. n. al. Ch. engchi U. v ni. 1 4. 2.1. Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.3. Expected waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. The probability of numbers of customers in the queues. 8. 3.1. Analysis of high-priority customers . . . . . . . . . . . . . . . . . . . . . . .. 9. 3.2. Analysis of low-priority customers . . . . . . . . . . . . . . . . . . . . . . . .. 11. 3.2.1. Truncation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.2.2. Matrix-product rate . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. Method to compute the probability of all servers idle . . . . . . . . . . . . . .. 17. 3.3 4. iii. ‧. List of Tables. ii. 學. List of Figures. 3. 政治大. 要. er. 中文. i. Analysis of Queueing Delays. 21. iii.

(5) 5. 6. 4.1. Analysis of ModelFCFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 4.2. Analysis of Model LCFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. Numerical results. 28. 5.1. Comparison the probability of all servers idle . . . . . . . . . . . . . . . . . .. 28. 5.2. Comparison between FCFS and LCFS . . . . . . . . . . . . . . . . . . . . . .. 32. Conclusion. 37. Appendix. 政治大. A The probability of all server idles and LST of the virtual waiting time. 立. B Computation of R(n). 41. ‧ 國. 學 42. ‧. io. sit. y. Nat. n. al. er. Bibliography. 38. Ch. engchi. iv. i n U. v.

(6) List of Figures 2.1. A model of two classes impatient customers . . . . . . . . . . . . . . . . . . .. 5.1. Probability of all server idle with different service rate. . . . . . . . . . . . . .. 30. 5.2. . . . . . . . . .. 31. 5.3. Probability of all server idle with different class-1 arrival rate. . . . . . . . . .. 31. 5.4. Probability of all server idle with different queue size. . . . . . . . . . . . . . .. 32. 5.5. Expected waiting time given service and abandonment. (γ1 = γ2 = 0.5, λ1 =. 政治大 Probability of all server idle with different class-2 arrival rate. 立 ‧. ‧ 國. 學. λ2 = s/2, µ = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. sit. Expected waiting time given service and abandonment in Jouini and Roubos [7].. n. al. er. io. (γ1 = γ2 = 0.5, λ1 = λ2 = s/2, µ = 1) . . . . . . . . . . . . . . . . . . . . 5.7. i n U. v. Ch. engchi. 34. Expected waiting time given service and abandonment in Jouini and Roubos [7]. (γ1 = γ2 = 1, λ1 = λ2 = s/2, µ = 1) . . . . . . . . . . . . . . . . . . . . .. 5.9. 34. Expected waiting time given service and abandonment. (γ1 = γ2 = 1, λ1 = λ2 = s/2, µ = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.8. 33. y. Nat. 5.6. 5. 35. Expected waiting time given service and abandonment. (K = 15, ϵ = 10−10 , µ = 5, γ1 = 3, γ2 = 5, λ1 = 10, λ2 = 2s) . . . . . . . . . . . . . . . . . . . . . .. 35. 5.10 Expected waiting time given service and abandonment. (K = 15, ϵ = 10−10 , µ = 5, γ1 = 3, γ2 = 5, λ1 = 10, λ2 = 4s) . . . . . . . . . . . . . . . . . . . . . .. 36. 5.11 CPU time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. v.

(7) List of Tables Difference between P0 and p0 . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. 政治大. 學 ‧. ‧ 國 io. sit. y. Nat. n. al. er. 5.1. Ch. engchi. vi. i n U. v. 29.

(8) Chapter 1 Introduction 政治大 In this thesis, we consider a system of multi-server queueing models with two classes of 立. ‧ 國. 學. impatient customer, in which one class is given higher priority than the other. Our focus is on the non-preemptive priority policy related to the start of service, wherein service cannot be. ‧. interrupted by other customers and impatient customers are prone to abandoning the system. Priority queues and the issue of abandonment are encountered in many applications, such as. y. Nat. io. sit. telecommunication networks, customer contact centers, and healthcare systems. Customers join. n. al. er. the system according to a Poisson process, and customers may abandon service after waiting in. i n U. v. a queue for an exponentially distributed duration. This study deals with two common service. Ch. engchi. disciplines: last come - first served (LCFS) and first come - first served (FCFS). High priority customers are dealt with by focusing on performance measures related to queueing times and conditional waiting times in cases where service is provided and in cases of abandonment. The focus is on expected waiting times when dealing with low-priority customers. An approximation of stationary probability distribution is obtained using a direct truncation method. The proposed method also uses Laplace-Stieltjes transforms (LST) to measure the waiting time distribution. Baccelli and Hebuterne [1] presented analysis of a queue system in which impatient customers are prone to abandonment. Garnett et al. [4] presented the simplest abandonment model, in which patience is exponentially distributed among all customers and the waiting capacity of the system is unlimited. Brandt and Brandt [2] presented a multi-server queueing system wherein customers may leave due to impatience. Choi et al. [3] introduce a simple approach to 1.

(9) the analysis of M/M/c queues using a single class of customers and constant impatience time. This study included two classes of customer: class-1 (customers with impatience of constant duration) and class-2 (customers with patience and lower priority than class-1 customers). Other researchers have dealt with non-preemptive priority queue systems. Kella and Yechiali [10] used probabilistic equivalence between the M/G/1 queue with multiple server vacations and the M/M/c system, in which the Laplace-Stieltjes transform is applied to waiting times. Sleptchenko [15] developed a multi-class, multi-server queueing system with non-preemptive priorities, in which steady-state probabilities are estimated. Zeltyn et al. [17] introduced a multiserver queue with K priority classes. The LST of waiting times is calculated explicitly and the. 政治大. LST of sojourn times is provided in an implicit form via a system of functional equations. Choi. 立. et al. [3] analyzes M/M/1 queues with impatient customers of higher priority. Kao and Wil-. ‧ 國. 學. son [9] analyzed non-preemptive priority queues with multiple servers and two priority classes. They developed a multi-server queueing system with two priority classes, in which high priority. ‧. customers have non-preemptive priority over low priority customers.. sit. y. Nat. Wang [16] considered a single-server with non-preemptive priority queuing for two classes. io. er. of impatient customers. Iravani and Balcıog̃lu [5] analyzed three different problems in which one class of customer is given priority over another class. In the first problem, a single server receives. n. al. Ch. i n U. v. two classes of customers with general service time requirements and follows a preemptive-. engchi. resume policy. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Sarhangian and Balciog̃lu [14] analyzed three delay systems where different classes of impatient customers arrive according to independent Poisson processes. In all models, they obtain the LST of the virtual waiting time for each class by exploiting the level-crossing method. This enables us to compute the steady-state system performance measures. Jouini and Roubos [7] recently proposed multi-server queues with two classes of impatient customer: high-priority and low-priority. The two classes have different arrival rates but the same abandonment rates and service is performed according to LCFS and. 2.

(10) FCFS. They explicitly derive the LST of the defined random variables. They compare FCFS and LCFS and gain insights through numerical experiments. They have derived the expected waiting time of two classes of customers with the same impatience rate while Sarhangian and Balciog̃lu [14] considered the expected waiting time under only FCFS. The goal of thesis is to investigate the expected waiting time of both classes customers with different impatient rates under LCFS and FCFS. The method presented in this study can be summarized as follows. We assign an expression for the LST of the random variable related to busy periods in order to investigate high-priority customers in an LCFS queue. Our analysis of high-priority customers is based on the method. 政治大. proposed by Jouini et al. [8] which derives all moments of the probability distribution of con-. 立. ditional waiting time in cases where service is provided and in cases of abandonment. We also. ‧ 國. 學. adopted the method of Jouini and Roubos [7] wherein we derive the stationary probability of class-1 customers in queue-1. For low-priority customers, we obtain the stationary probability. ‧. of class-2 customers in queue-2 based on truncation and the matrix analytic method.. sit. y. Nat. This thesis is organized as follows. In Sections 2.1 and 2.2, we describe two classes of. io. er. queueing model with impatient customers, and define the notation in this thesis. In Section 2.3, we outline some of the basics pertaining to expected waiting times. In Sections 3.1 and 3.2,. n. al. Ch. i n U. v. we outline a method by which to compute stationary probability for the number of customers. engchi. in different queues. Section 3.3 proposes a method by which to compute the probability of all servers being idle at the same time. In Section 4.1, we analyze the expected waiting time in cases where service is provided and in cases of abandonment for high- and low-priority customers under conditions of FCFS. In Section 4.2, we analyze the expected waiting times in cases where service is provided and in cases of abandonment for high-priority customers, and analyze the expected waiting time for low-priority customers. In Section 5, we present numerical results. In Section 6, we draw conclusions and indicate directions for future research.. 3.

(11) Chapter 2 Preliminaries 立. Modeling. 學. ‧ 國. 2.1. 政治大. Consider a queueing model with two types of customer: important (high-priority) customers. ‧. denoted as class-1, and less important (low-priority) customers denoted as class-2. In the following, we model a queueing system in which identical multi-servers attend to the. y. Nat. io. sit. two classes of impatient customer. We adhere to the non-preemptive rule, in which a server can. n. al. er. attend to a class-2 customer only when there is no class-1 customer in the queue; and class-1. i n U. v. customers must wait for the completion of service for class-2 customers before being served.. Ch. engchi. The proposed model in Figure 2.1 comprises a finite-buffer queue for class-1 customers (queue-1), an infinite buffer for class-2 customers (queue-2), as well as a set of s identical servers running in parallel. In cases where all servers are busy, newly arriving class-1 customers must wait in queue 1 and newly arriving class-2 customers must wait in queue-2. All of the servers provide identical service rates to both types of customer and the system is conservative (i.e., no server is forced to be idle when customers are waiting). In the following, we consider two cases of service: FCFS and LCFS. Class-i customers arrive at the queueing system according to an independent Poisson process at the following rate: λi , i=1,2. The total arrival rate is denoted by λ = λ1 + λ2 Both classes of customer are assumed to be impatient, as follows. After entering a queue in which all servers are busy, each arriving customer waits for a random length of time. If the 4.

(12) service time exceeds the waiting time, the customer abandons the queue. All customers are assumed to be lacking in memory of the previous queuing experience; therefore, when service begins, they must wait again until the service is completed. We assume that class-i customers have exponentially distributed time-to-abandon at rate γi , i=1,2. We denote by EX k the k-th order moment of a given random variable X, for k ≥ 1. We also denote by f X (·) and FX (·) the probability density function and the cumulative distribution function of X. Furthermore, we know that the 1st order moment of a given random variable X is the expected value of X.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.1: A model of two classes impatient customers. 2.2. Notation. 5.

(13) W:. a random variable, the unconditional total waiting time of an arbitrary customer in the queue.. Wi :. a random variable, the unconditional waiting time of a class-i customer in the queue.. Wi,s :. a random variable, the conditional waiting time of a class-i customer, given that he will enter the service.. pi,s :. the probability that a class-i customer in the service.. Wi,a :. the conditional waiting time of a class-i customer, given that he will abandon the service.. 政治大. pi,a :. the probability that that a class-i customer abandons the service.. Wi,w :. a random variable, the conditional waiting time of a class-i customer,. 學. ‧ 國. 立. given that he has to wait. the probability that a new arrival has to wait.. Wi,w,s :. a random variable, the conditional waiting time of a class-i customer,. sit. Nat. given that he has to wait, and he will enter service.. y. ‧. pw :. Qi :. a random variable, the unconditional numbers of class-i customers in the queue.. er. the probability that a new arrival has to wait and he will enter service.. io. pi,w,s :. al. n. v i n C hwill necessarily enter A customer who does not abandon e n g c h i U the service, namely, we have pi,s +. pi,a = 1. A new customer who waits in the queue has two choices: He either enters the service or abandons in the queue, thereby implying that pi,w,s + pi,a = pw .. 2.3. Expected waiting time. Because the arrival process is Poisson process, the probability of the new customer being class i is λi /λ, where λ = λ1 + λ2 . So, we have. EW =. λ2 λ1 EW1 + EW2 . λ λ. 6.

(14) Because of pi,s + pi,a = 1, we obtain. EWi = pi,s EWi,s + pi,a EWi,a .. When the new customer arrivals, he may either enter the service immediately or wait in the queue. So. EWi = pw EWi,w .. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 7. i n U. v.

(15) Chapter 3 The probability of numbers of customers in the queues 政治大立 ‧ 國. 學. Denoted by n1 (t), n2 (t), and n(t) the number of customers in the queue-1, the number of. y. Nat. n1 ( t ) + n2 ( t ).. ‧. customers in queue-2, and the total number of customers in both queues respectively, n(t) =. n. al. er. io. denoted by K.. sit. We assume that the maximum number of customers in the finite-buffer queue for class-1 is. i n U. v. In the long run, let (n1 , n2 ) represent the system state and their possible transitions be given. Ch. engchi. in the following table. Notice that γ1 ̸= γ2 .. 8.

(16) Transition Rates From. To. Rate. Condition 0 ≤ n1 ≤ K − 1,. (n1 , n2 ) (n1 + 1, n2 ) λ1. & n2 ≥ 0 0 ≤ n1 ≤ K,. ( n1 , n2 ) ( n1 , n2 + 1) λ2. & n2 ≥ 0. (n1 , n2 ) (n1 − 1, n2 ) s · µ + n1 · γ1 1 ≤ n1 ≤ K, & n2 ≥ 0. 政治大 &n. (n1 , n2 ) (n1 , n2 − 1) s · µ + n2 · γ2 n1 = 0,. 立. ‧ 國. & n2 ∈ N. Analysis of high-priority customers. Nat. y. ‧. 3.1. n1 ̸= 0,. 學. (n1 , n2 ) (n1 , n2 − 1) n2 · γ2. ∈N. 2. io. sit. Borrowing the terminology from Jouini and Roubos [7], we have the stationary probability. n. al. er. of number of high-priority customers in this chapter. The stationary probability of number of. i n U. customers in the service is denoted by pk , 1 ≤ k ≤ s − 1. Thus pw is given by. Ch. engchi. pw = 1 −. v. s −1. ∑ pk .. k =0. The normalization condition gives s −1. K. ∞. ∑ pk + ∑ ∑ πk,j = 1,. k =0. k =0 j =0. where πk,j is the stationary probability, k class-1 customers are in queue-1, and j class-2 customers are in queue-2, when all servers are busy. Let π1 (k) denote the stationary probability that all servers busy and there are k class-1 cus-. 9.

(17) tomers in queue-1. Summing up all the states with respect to k, this system corresponding the M/M/s+M queue. The equation for class-1 leads to λ1k. π1 ( k ) =. ∏kj=1 (sµ + jγ1 ). π1 (0),. And we know that ∞. ∑ πk,j ,. π1 ( k ) =. j =0. 政治大. Applying π1 (k ) to normalization condition we have. 立∑p. K. k+. k =0. ∑ π1 (k) = 1,. k =0. 學. ‧ 國. s −1. Also, it can be written as λ1k. k =0. ∏kj=1 (sµ + jγ1 ). ∑. π1 (0) = 1,. y. k =0. K. io. sit. Nat. ∑. pk +. ‧. s −1. n. al. er. By the above equation we obtain. C sh−1 K ∑epkn)(g∑c hk i. π1 (0) = (1 −. k =0. k =0. i n k Uλ. v. 1. ∏ j=1 (sµ + jγ1 ). ) −1 ,. Therefore we have the expected number of customers in queue-1 which is given by K. EQ1 =. ∑ kπ1 (k). k =1. Because the queue-1 is finite, we have the blocking probability pb as follows: s −1. K −1. k =0. k =0. pb = 1 − ( ∑ pk +. 10. ∑ π1 (k))..

(18) From Sarhangian and Balciog̃lu [14], pi,s is given by pi,s = 1 − pi,a = p0 +. ∫ ∞ 0. e−γi y f i (y)dy = p0 + f˜i (γi ),. where f i ( x ) denoted the density function of the virtual waiting time for class i customers and f˜i (s) denoted the LST’s of f i ( x ). By Eq. (13) in Sarhangian and Balciog̃lu [14], the waiting time distribution of class i customer in the queue is P(Wi ≤ x ) = 1 − e−γi x + e−γi x Fi ( x ),. 政治大. 立. 學. ‧ 國. where Fi ( x ) denoted the cumulative distribution function of the virtual waiting time for class i customers. And it has the expected value. 0. e−γi x F¯i ( x )dx. n. sit er. io. al. y. p 1 − f˜i (γi ) − p0 = = i,a . γi γi. Nat So, we have. 0. P(Wi > x )dx =. ∫ ∞. ‧. EWi =. ∫ ∞. Ch. eng c hi i γi EQ. pi,a =. λi. i n U. v. ,. pi,s = 1 − pi,a , i = 1, 2.. 3.2. Analysis of low-priority customers. Let πk,j denote the stationary distribution of {n1 (t), n2 (t); 0 ≤ n1 (t) ≤ K, n2 (t) ≥ 0, t ≥ 0}, i.e, πk,j = lim P[n1 (t) = k, n2 (t) = j] t→∞. 11.

(19) Based on the classification of the states, we derive the infinitesimal generator matrix Q for the QBD process as follows: . .  Q1 Q2  Q= , Q3 Q4 We also define pk , k = 0, 1, 2, · · · , s − 1 be the probability that k customers is served, and denote p = ( p0 , p1 , · · · , ps−1 ). We then have. 立. ( p, π )Q治 = 0, 政大 ( p, π )1T = 1,. ‧ 國. 學. also it can be written as. ‧. π1T = pw ,. sit. y. Nat. n. al. er. io. where 0 and 1T denote a row vector of zeros and column vector of ones. And Qk , k = 1, 2, 3, 4 can be shown as follows:   −λ λ 0 ··· ··· ··· 0 0  µ −(λ + µ) λ 0 · · · · · ·  0 0    ..  . . . . . . . . . . . . . . Q1 =  . , . . . . . . .    0  0 · · · · · · · · · · · · − λ − ( s − 2) µ λ 0 0 ··· ··· ··· ··· ( s − 1) µ − λ − ( s − 1) µ where Q1 is s × s matrix. Ch. engchi.       Q2 =      . i n U. 0 0 ··· 0 0 0 ··· .. . . . . . . . 0 0 ··· E 0 ···. 12. .   0  ..  . ,   0  0. v.

(20) .  0  0  Q3 =  .  ..   0 . B0 A1 0 · · · · · · · · · · · · 0.    C0   Q4 =   0    0  .. .. 立. ··· 0 D   ··· 0 0   , . . .. ..  . . .   ··· 0 0. B1 A2 0 · · · · · · · · · C1 B2 A3 0 · · · · · · 0 C2 B3 A4 0 · · · ... ... ... ... ... .... 政治大. .   0   0 ,   0  .. .. 學. ‧ 國. Each elements in Qk , for k = 2, 3, 4 is defined as follows: . n. and D is the (K + 1) × 1 column matrix.. Ch. y. sit er. io. al. ‧. Nat.  sµ      0    D =  . ,  .   .    0. e[n g c h i. i n U ]. λ 0 ··· 0. E=. v. ,. and E is the 1 × (K + 1) row matrix.       Aj =      . λ2 0 · · · · · · 0 0 λ2 0 · · · 0 0 .. .. 0 λ2 · · · .. . . . . . . .. 0. 0 · · · · · · λ2. 0 .. ..       ,     . where A j is (K + 1) × (K + 1) matrices for all j = 1, 2, · · · .. 13.

(21) For brevity, we use the following simplified notation. Uj,k = −(sµ + kγ1 + jγ2 + λ1 + λ2 ), ∀ j = 0, 1, 2, . . . , ∀ k = 0, 1, 2, . . . , K − 1, Uj,K = −(sµ + Kγ1 + jγ2 + λ2 ), ∀ j = 0, 1, 2, . . . ,. and Tk = sµ + kγ1 , ∀ j = 1, 2, . . . , K,. (3.2.1). Then we have B j and C j written as follows :. Nat. 0 .. . .. . .. .. io. n. a0 l. . λ1. T2 Uj,2 ... 0 .. .. . . .. .. . . .. .. . . 0. Ch. 0 ··· ···. ···. λ1 0 · · · ··· ... ... ... ... .. .. .. .. . . . . .. .. .. .. . . . . .. .. . . TK −1 Uj,K −1. y. 0. j,1.   0    0    ..  .  , ..  .    0     λ1   Uj,K. sit. 1.  0. ‧. ‧ 國. 立 T U. 政0 · · ·治 ··· ··· 大 ···. 學.           Bj =           . Uj,0 λ1. ··· ··· ···. 0. er. . TK. iv. n U i e n0 g c h · · · · · ·.  sµ + ( j + 1)γ2   0 ( j + 1)γ2 0    0 0 ( j + 1)γ2  Cj =   .. .. ..  . . .   .. .. ...  . .   0 0 ···. . ···. 0. ··· ···. 0. 0 ··· .. .. . . ... .... 0 .. . 0. · · · 0 ( j + 1)γ2.        ,       . where B j and C j are (K + 1) × (K + 1) square matrices. We have the state (k, j) for 0 ≤ k ≤ K and j ≥ 0, i.e., (0, 0), (1, 0), · · · , (K, 0), (0, 1), (1, 1), · · · . Let π j and π denote π j = (π0,j , π1,j , · · · , πK,j ) and π = (π 0 ,π 1 , · · · , π j , · · · ).. 14.

(22) Moreover, according to the matrix analytic method, π j has a matrix-product form solution given by j. π j = π 0 ∏ R(k) , j ≥ 1. k =1. The vector π 0 satisfies the following equations: ps−2 λ + ps−1 [−λ + (s − 1)µ] + π 0 D = 0, ps−1 E + π 0 B0 + π 1 C0 = 0.. Also, it can be written as. 立. 政治大. ‧ 國. 學. π 0 D = − ps−2 λ − ps−1 [−λ + (s − 1)µ],. ‧. π 0 (B0 + R(1) C0 ) = − ps−1 E,. sit. y. Nat. given ps−1 , ps−2 and R(1) we can obtain the unique solution of π0 .. n. al. An + R. (n). er. io. And { R(n) ; n ≥ 1} is the minimal nonnegative solution to the following system of equations:. v i n C Bn + R R h e n g c hCin =UO, n ≥ 1, ( n ) ( n +1). where O is the matrix which each element is zeros. So, if we obtain R(n) for all n, then we have π j for j ≥ 1 .. 3.2.1 Truncation point In the previous chapter, the stationary distribution π n of our QBD process can be expressed by a matrix-product rate R(n) . But R(n) has no explicit form. So, we can only compute π¯ n as the approximate of π n , for 0 ≤ n ≤ N, where N is the truncation point of π n . We advance a method to compute N. For this truncation point N, we can obtain an approximate π¯ n . We need to choose a number N by which the tail probability π¯ N is small enough to. 15.

(23) be neglected. Because of our model is similar to an M/M/s queue, we can use the stationary probability expression in M/M/s model to find N, such that the tail probability is small enough. Our model has arriving rate λ1 and λ2 , abandon rate γ1 and γ2 , and the maximum service rate s × µ. Let λ¯ = λ1 + λ2 − γ1 − γ2 ,. ¯ s, µ in an M/M/s queue. by using λ,. 立. n. N = inf{n | 1 − ∑ p¯ i < ϵ}. 學. ‧ 國. We have. 政治大 i =0. ‧. where p¯ n is the stationary probability of M/M/s queue, and ϵ we can chosen small enough.. sit. y. Nat. n. al. er. io. 3.2.2 Matrix-product rate. i n U. v. When the truncation point N is given, the stationary probability distribution can be obtained.. Ch. engchi. Moreover, because of N is chosen sufficiently large, so the probability after N is small enough that we can disregard. In this thesis, we use a method to compute Rn in Proposition 1 and 2. Proposition 1 is borrowed from Proposition 1 in Phung-Duc et al. [13] and proposition 2 is borrowed from Proposition 2.4 in Phung-Duc et al. [12]. Proposition 1. Let S denote a set of real square matrices of ( N + 1) × ( N + 1). We define Gn : S → S as Gn ( X ) = An (−Bn − XCn )−1 , n ≥ 1.. 16.

(24) Then, the matrices R(n) for n ≥ 1 satisfies the following backward recursive equation. R(n) = Gn ( R(n+1) ) = Gn ◦ Gn+1 ◦ · · · ◦ Gn+k ◦ · · · , n ≥ 1, where f ◦ g(·) = f ( g(·)) is the composition function.. This proposition tells that R(n) can be obtained for an infinite matrix composite. And the next proposition will provide a method that converges to R(n) . (n). Proposition 2. If we define the matrix Gk. for n ≥ 0 and k ≥ 0 by. 政治大. 立. = O, k = 0,. (n). = Gn ( Gk−1 ) = · · · = Gn ◦ Gn+1 ◦ · · · ◦ Gn+k−1 (O), k ≥ 1, n ≥ 1,. n. (n). er. io. al. The proposition means that Gk. = R(n) , n ≥ 1.. v. is the k-th order approximate of R(n) . It also tells that we. Ch. engchi. can obtain R(n) sufficiently close by a fairly large k.. 3.3. sit. Nat. (n). lim Gk. k→∞. y. ‧. then we have. ( n +1). 學. Gk. ‧ 國. (n). G0. i n U. Method to compute the probability of all servers idle. As the last chapter, the probability of pk can be given by M/M/s queue :. pk =. λk p0 , 0 ≤ k ≤ s − 1, k!µk. where p0 is the probability of all servers idle. In order to compute pk , we need a method to calculate p0 . Moreover, we propose a proposition to compute p0 . 17.

(25) Proposition 3. If we define the probability x0 by. x0 = 1, xk =. λk x0 , 0 ≤ k ≤ s − 1, k!µk k. xs+k =. ∏ R( j) xs ,. k = 1, 2, · · · , N. j =1. and define. ( x0 , x1 , · · · , xs−1 , x)Q = 0,. 立. where x denote x = ( xs ,xs+1 , · · · ).. s −1. p0 = ( ∑ x k + k =0. N. ∑ x s + k e ) −1 .. ‧. ‧ 國. 學. Then we have. 政治大. k =0. Nat. y. (3.3.1). n. al. x0 = tp0 = 1,. Ch. engchi. er. io. sit. Proof. First, we know that. i n U. v. where p0 is define before. Also, it can be written as p 0 = t −1 . Second, we need to prove that xk = tpk , k = 0, 1, 2, · · · , s − 1 and xs+k = tπ k , k = 0, 1, 2, · · · , K. Clearly, because of. xk =. λk x0 , 0 ≤ k ≤ s − 1, k!µk. 18.

(26) then we have λk tp0 = k!µk. xk. = t · pk , for all k = 0, 1, · · · , s − 1. The vector xs is a solution from the equations xs−2 λ + xs−1 [−λ + (s − 1)µ] + xs D = 0, x E +治 x B +x C 政大 s −1. 立. Also, it can be written as. s 0. s +1. 0. = 0.. ‧ 國. 學 xs (B0 + R(1) C0 ) = t(− ps−1 E),. er. io. sit. y. Nat. therefore we have. ‧. x0,0 sµ = t{− ps−2 λ − ps−1 [−λ + (s − 1)µ]}. al. n. v i n C hps−2 λ − ps−1 [−λU+ (s − 1)µ]}/sµ = t{− engchi. x0,0. = tπ0,0 , xs. = t(− ps−1 E)(B0 + R(1) C0 )−1 = tπ 0 ,. it means that xs = tπ 0 . By the matrix-product forms k. xs+k = xs. ∏ R(n) ,. n =1. 19. k ≥ 1,.

(27) then we have k. xs+k = tπ 0. ∏ R(n). n =1. = tπ k , k ≥ 1. Finally, we obtain s −1. ∑. N. xk +. k =0 s −1. ∑ xs+k. k =0 N. = t{ ∑ pk +. ∑π } 治政大 = t×1 k =0. 立. = t.. ‧. ‧ 國. 學. io. sit. y. Nat. p 0 = t −1. n. al. s −1. = ( ∑ xk +. Ch. N. er. Moreover,. k. k =0. ∑ xs+k )i−v1.. n U i engch k =0. 20. k =0.

(28) Chapter 4 Analysis of Queueing Delays 政治大 For n ≥ 1, an n-busy period is defined as the any point of time from the arriving customer to 立. ‧ 國. 學. a busy M/M/s + M system with n − 1 waiting customers in the queue until the time at which one server becomes idle.. ‧. Moreover, the 0-busy period is the classical busy-period definition defined to begin with the. y. sit. again.. Nat. arriving customer to a system with s − 1 busy servers and end with one server becomes idle. n. al. er. io. Suppose there is only one class of customers where the arrival rate is λ, the abandonment rate. i n U. v. γ and the service rate is µ. We denoted the length of an n-busy period by Bn,λ,γ , for n ≥ 1. The. Ch. engchi. Laplace-Stieltjes transform of the pdf of Bn,λ,γ by F˜Bn,λ,γ ( x ), is found from Eq. (1) of Iravani and Balcıog̃lu [5] by substituting b˜ 1 ( x ) = sµ/( x + sµ). We obtain. F˜Bn,λ,γ ( x ) =. sµ x +sµ. 1 + ∑i∞=1 (−1)i [∏ij− =0 ( 1 −. sµ sµ x +sµ+ jγ )] x +sµ+iγ Θ ( n, i ) , sµ i −1 λi ( 1 − 1 + ∑i∞=1 i!γ [ )] ∏ i x +sµ+ jγ j =0. with  j j   ∑i (−1) j λ ( n ), 1 ≤ i ≤ n, j=0 j!γ i− j Θ(n, i ) = j j   ∑ij=i−n (−1) j λ ( n ), i > n, i− j j!γ. 21.

(29) And the 0-busy period in finite queue is given by. F˜B0,λ,γ ( x ) =. sµ x +sµ. i −1 λ + ∑iK=1 i!γ i [ ∏ j =0 ( 1 − i. sµ sµ x +sµ+ jγ )] x +sµ+iγ , sµ i −1 λi 1 + ∑iK=1 i!γ i [ ∏ j =0 (1 − x + sµ + jγ )]. In section 4.2, because our analyze the performance of class-1 that have γ1 abandon rate and λ1 arrival rate, we assume that γ = γ1 and λ = λ1 .. 4.1. Analysis of ModelFCFS. 政治大 customers given served and abandon, 立 and also compute the expected waiting time in the queue For high and low priority customers, we focus on the conditional probability of high-priority. ‧ 國. 學. for low-priority customers.. ‧. High-Priority Customers. y. Nat. waiting which it is 1-st moment of W1,s and W1,a .. n. al. Ch. er. io. sit. In the follows, we use the k-th order moment of W1,s and W1,a to compute the expected. i n U. v. Consider the new class-1 customer arrival the system, and he find all servers busy, and n1 waiting. engchi. customers ahead in queue-1. And the contrary case (at least one server idle), he will immediately enter the service. The class-2 customers, because of their low-priority, will not affect the class-1 customers sojourn time. Using Jouini et al. [8], Jouini and Roubos [7], we obtain k = EW1,s. 1 p1,s. K. ∑. n1 =0. p1 (n1 )Ψn1 +1 EYnk1 +1 ,. with Ψ n1 =. n1. γ1. sµ. ∏(1 − sµ + iγ1 ) = sµ + n1 γ1 . i =1. 22. (4.1.1).

(30) Yn1 , a random variable, is the summation of n1 independent exponential distributions with parameters sµ + γ1 , sµ + 2γ1 , · · · , sµ + n1 γ1 . Its first moment is n1. EYn1 =. 1. ∑ sµ + jγ1 ,. j =1. Considering the EW1,a , the new class-1 customer arrival who find at least one server idle, the expected conditional waiting time of class one customer given abandon. Let Zn1 +1 denote the random variable measuring her sojourn time in the queue before abandonment. Removing the condition on n1 , we obtain. 治政 1 大 = ∑ p (n )EZ K. 立. EW1,a. P1,r. 1. 1. n1 +1 .. (4.1.2). n1 =0. ‧ 國. 學. Seeing the probability to abandon at position j, for 1 ≤ j ≤ n1 , we obtain n1. ∏. (1 −. sit. y. Nat. l = j +1. γ1 γ1 )= . sµ + lγ1 sµ + n1 γ1. ‧. γ1 sµ + jγ1. io. n. al. er. Averaging over all possibilities, we have γ1 sµ + n1 γ1. Ch. EZn1 =. n1. i n U. v. ∑ EZn1 ( j).. e n g c jh=1i. The expected value of Zn1 can be written as EZn1. γ1 = sµ + n1 γ1. n1. jγ1. ∑ sµ + jγ1 .. j =1. Low-Priority Customers. For a new type 2 customer, we only focus on the expected waiting time. In the last few chapter, we have the approximate of πn . Having π n on hand, we can compute the expected queueing length for class-2 customer. There-. 23.

(31) fore, we obtain ∞. EQ2 =. ∑ i · πi · 1T ,. i =1. where 1T is the column vector with element 1. Also, because of the class-2 has infinite queue, so we have. EW2 =. EQ2 . λ2. 政∫ 治大 xe f ( x )dx. From the Eq.(11) in Sarhangian and Balciog̃lu [14], we have. 立EW. =. −γ2 x. 2. .. (4.1.3). 學. pi,s. ‧ 國. 2,s. ∞ 0. Having the LST of f i ( x ) Eq. (A.0.2) on hand, Eq. (4.1.3) can be compute by taking the derivate. ‧. with respect to γ2 .. EW2 = p2,s EW2,s + p2,a EW2,a .. n. al. er. io. sit. y. Nat. Also, EW2,a can be obtained by. 4.2. Ch. engchi. i n U. v. Analysis of Model LCFS. High-Priority Customers. Approaching to compute the expected waiting time is base on their virtual waiting time. The virtual waiting time is defined as the waiting time given that the customer has infinitely patient. Let Vi (t) be the virtual waiting time of a class-i customer at time t with f i ( x ) as its density function. We define the patience times by the random variable T. We focus on the conditional waiting time of class-1 customer given served, and we obtain. FW1,s (t) =. P(V1 < t, V1 < T ) , P(V1 < T ) 24.

(32) for t ≥ 0. Because of the discipline of service is LCFS, class-1 customer already in the queue is not affect the waiting time. So the conditional virtual waiting time given that the new customer has to wait is independent of the state, and denote it by V1,w . We have V1 = pw · V1,w. Also, we have. 1. 1,s. 學. and. ∫ t. P(V1 < t, V1 < T ) =. 0. ‧. ‧ 國. 立. 政治大 P (V < T ) = p ,. e−γ1 x f 1 ( x )dx.. sit. y. Nat. io. with probability 1 − pw . Or, he finds all server busy.. n. al. Ch. er. Therefore, a new class-1 customer finds at least one idle server will immediately enter the service. i n U. v. Because of the conditional virtual waiting time is independent of the state. Thus we can write. engchi. P(V1 < t, V1 < T ) = (1 − pw ) · 1 + pw. ∫ t 0. e−γ1 x f V1,w ( x )dx.. We can see that V1,w is equivalent to the 0-busy period in M/M/s+M queue. So, we have 1 {1 − p w + p w FW1,s (t) = p1,s. ∫ t 0. e−γ1 x f V1,w ( x )dx },. Taking the derivative with respect to t above, we obtain. f W1,s (t) =. pw −γ1 t e f B0,λ ,γ (t), 1 1 p1,s 25.

(33) Using the LST we gain pw ˜ F˜W1,s (t) = FB ( x + γ1 ), p1,s 0,λ1 ,γ1 Applying the 1-st order moment EW1,s = −. p w ˜ (1) F (γ1 ), p1,s B0,λ1 ,γ1. (4.2.1). where g(k) (·) is the k-th derivative of g(·).. 政治大. Now, we focus on the conditional waiting time given abandonment. Having that. ‧ 國. FW1,a (t) =. P( T < t, V1 > T ) , P(V1 > T ). 學. 0. γ1 e−γ1 x (1 − Pw + Pw FB0,λ. 1 ,γ1. ( x ))dx },. n. er. io. al. ∫ t. sit. Nat. 1 FW1,a (t) = {1 − e−γ1 t − p1,a. y. ‧. We obtain. 立. Taking derivative with respect to t on both side. Ch. f W1,a (t) =. engchi. i n U. v. Pw γ1 −γ1 t (e − e−γ1 t FB0,λ ,γ (t)), 1 1 p1,a. Applying the LST, we obtain F˜W1,a ( x ) =. Pw γ1 (1 − F˜B0,λ ,γ ( x + γ1 )), 1 1 P1,w ( x + γ1 ). (4.2.2). Finally, taking the k-th order moment of W1,a , we obtain the expected conditional waiting time given abandonment.. Low-Priority Customers. 26.

(34) Using Jouini and Roubos [7], we know that the expected waiting time is unchanged for all work-conserving policies. Clearly, because the Markov chain is unchanged, so does the matrix Q. Then we know that π n is same as before. Having π n on hand, we can compute the expected queueing length for class-2 customer. Therefore, we obtain ∞. EQ2 =. ∑ i · π i · 1T ,. i =1. where 1T is the column vector with element 1. Also, because of the class-2 has an infinite queue, so we have. 立. 政治 EQ . 大 EW = 2. 2. λ2. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 27. i n U. v.

(35) Chapter 5 Numerical results 政治大 In this chapter, we use numerical solutions to prove that our results are reasonable. First, we 立. ‧ 國. 學. use the same conditions to state that our solution is the same as the solution given in Eq. (3.3.1). Second, we use Eqs.(4.1.1), (4.1.2), (4.2.1) and (4.2.2) to gain the useful result.. ‧. Comparison the probability of all servers idle. sit. y. Nat. 5.1. n. al. er. io. In the Sarhangian and Balciog̃lu [14], it gives s −1. P0 = ( ∑. i =0. λi. i!µi. Cλh. +. λ c −1. ∞. i n U j. v. ∏ λ2 g˜0 (kγ2 ))−1 e n g c h ij∑ =0. λ2 (c − 1)!µc−1. k =0. where g˜0 ( x ) is the LST of B¯ 0,λ1 ,γ1 with the complementary distribution function of an 0-busy period. We make a comparison between this P0 and p0 in Eq. (3.3.1). Assuming λ1 = 30, λ2 = 40, s = 5, µ = 15, K = 15 , ϵ = 10−10 . Table 5.1 indicates that the difference between p0 and P0 is less than 10−4 , thereby demonstrating the accuracy of the proposed method. Moreover, an increase in the rate of abandonment leads to a decrease in the number of customers in the system, which in turn increases the probability of all servers being idle. This is a clear demonstration that these numerical results are reasonable. Next, we focus on differences in service rates using a graph. It is clear that an increase in. 28.

(36) Table 5.1: p0 γ1 0.0075 6 0.0077 0.0079 0.0081 0.0083 0.0084 0.0085 0.0087 0.0088 0.0089 0.0090 0.0077 10 0.0079 0.0081 0.0083 0.0085 0.0086 0.0087 0.0088 0.0089 0.0090 0.0091. n. Ch. sit. y. ‧. io. al. 政治大. er. 立. Difference between P0 and p0 γ2 P0 p0 γ1 γ2 5 0.0076 0.0076 7 5 6 0.0078 0.0078 6 7 0.0080 0.0080 7 8 0.0082 0.0082 8 9 0.0083 0.0083 9 10 0.0085 0.0085 10 11 0.0086 0.0086 11 12 0.0087 0.0087 12 13 0.0088 0.0088 13 14 0.0089 0.0089 14 15 0.0090 0.0090 15 5 0.0078 0.0078 15 5 6 0.0081 0.0081 6 7 0.0082 0.0082 7 8 0.0084 0.0084 8 9 0.0086 0.0086 9 10 0.0087 0.0087 10 11 0.0088 0.0088 11 12 0.0089 0.0089 12 13 0.0090 0.0090 13 14 0.0091 0.0091 14 15 0.0092 0.0092 15. 學. P0 0.0075 0.0077 0.0079 0.0081 0.0083 0.0084 0.0085 0.0087 0.0088 0.0089 0.0090 0.0077 0.0079 0.0081 0.0083 0.0085 0.0086 0.0087 0.0088 0.0089 0.0090 0.0091. Nat. 8. γ2 5 6 7 8 9 10 11 12 13 14 15 5 6 7 8 9 10 11 12 13 14 15. ‧ 國. γ1 5. engchi. 29. i n U. v. P0 0.0076 0.0079 0.0081 0.0082 0.0084 0.0085 0.0087 0.0088 0.0089 0.0090 0.0091 0.0081 0.0083 0.0085 0.0086 0.0088 0.0089 0.0090 0.0091 0.0092 0.0093 0.0094. p0 0.0076 0.0079 0.0081 0.0082 0.0084 0.0085 0.0087 0.0088 0.0089 0.0090 0.0091 0.0081 0.0083 0.0085 0.0086 0.0088 0.0089 0.0090 0.0091 0.0092 0.0093 0.0094.

(37) 0.25. probability of all server idle p0. 0.2. 0.15. 0.1. 0.05. 0 15. 立. 25. 30 35 service rate µ. 40. 45. 50. 學. ‧ 國. 20. 政治大. Figure 5.1: Probability of all server idle with different service rate.. ‧. the service rate leads to a reduction in the number of customers in the system, which increases. y. Nat. n. al. er. io. the probability of all servers being idle also increases.. sit. the probability of all servers being idle. Figure 5.1 shows that when the service rate increases,. i n U. v. We assume λ1 = 30, s = 5, µ = 15, K = 15 , ϵ = 10−10 , γ1 = 10, γ2 = 15 with different class-2. Ch. engchi. customer arrival rates, as shown in Figure 5.2. Therefore, it is easy to show that the probability of system idle decreases as the class-2 arrival rate increases. Thus, the probability of all servers being idle drops with an increase in the arrival rate. We assume that λ2 = 30, s = 5, µ = 15, K = 15 , ϵ = 10−10 , γ1 = 10, γ2 = 15 with different class-1 arrival rates, as shown in Figure 5.3. This is the same as the situation in Figure 5.2, because the probability of all servers being idle decreases when the arrival rate increases. In Figure 5.4, we consider differences in queueing size for high-priority customers with λ1 = 30, λ2 = 40, s = 5, µ = 15 , ϵ = 10−10 , γ1 = 10, γ2 = 15. Clearly, an increase in queue size of class-1 decreases the probability that all servers are idle in the system.. 30.

(38) 0.04. probability of all server idle p 0. 0.035 0.03 0.025 0.02 0.015 0.01 0.005. 立. ‧ 國. 25. 30. 35. 40. class-2 arrival rate λ2. 學. 0 20. 政治大 45. 50. n. al. er. io. sit. y. Nat. 0.04. ‧. Figure 5.2: Probability of all server idle with different class-2 arrival rate.. probability of all server idle p 0. 0.035 0.03. Ch. engchi. i n U. v. 0.025 0.02 0.015 0.01 0.005 0 20. 25. 30. 35. 40. 45. 50. class-1 arrival rate λ1. Figure 5.3: Probability of all server idle with different class-1 arrival rate.. 31.

(39) 9.9. ×10 -3. probability of all server idle p 0. 9.8. 9.7. 9.6. 9.5. 9.4. 政治大. 9.3. 立. 9.2 2. 4. 6. 8. 10. 12. class-1 queue size K. 14. 16. 18. 20. 學. ‧ 國. 0. Figure 5.4: Probability of all server idle with different queue size.. ‧. Comparison between FCFS and LCFS. sit. y. Nat. 5.2. er. io. The number of servers s is used as a variable in the numerical tests to determine the expected. al. v i n times of class-1 given service andCabandonment U in the same class in Figures 5.5 h e n g c ofh icustomers n. waiting times associated with two service policies. It depicts the conditional expected waiting. - 5.10. We begin by assigning the same rate of abandonment for both classes of customer, and compare the expected waiting time computed by Jouini and Roubos [7]. The model presented by Jouini and Roubos [7] assumes an infinite buffer for class-1 customers; however, we can still perform a comparison with the proposed model if we permit a sufficiently large buffer size for class-1 customers. Second, we assume that the rate of abandonment differs between the two classes of customers. Comparing with Figures 5.6 and 5.8 (borrowed form Jouini and Roubos [7]), we take Eqs.(4.1.1), (4.1.2), (4.2.1) and (4.2.2) to compute the expected waiting time in order to verify our model that are shown in Figures 5.5 and 5.7. Figure 5.5 shows that our numerical results are close to those presented by Jouini and Roubos [7]; therefore, it would be reasonable to describe the proposed model and computer code as valid. The error that does exist. 32.

(40) can be attributed to the difference in the size of the buffer queue for class-1 customers. For the single-class M/M/s+M queue, the conditional expected waiting for those given service in FCFS is greater than that under LCFS policy that was demonstrated by Jouini et al. [8]. Also, the conditional waiting time for those who abandon the queue in LCFS is greater than FCFS. These results are easily extended to our models in Figure 5.9 and Figure 5.10, wherein a doubling in the arrival rate affects only the conditional expected waiting time and the property does not change. We present the Central Processing Unit (CPU) time for the expected waiting time with different s in Figure 5.10. The computing times under FCFS and LCFS are significantly different. This is due to using Eqs.(4.1.1) and (4.1.2) for FCFS while using Eqs.(4.2.1) and (4.2.2) for LCFS.. 立. 政治大. ‧ 國. 學. 0.7. EW 1,a ,FCFS EW 1,s ,FCFS EW 1,a ,LCFS. ‧. EW 1,s ,LCFS. 0.5. y. sit. io. 0.3. n. al. er. 0.4. Nat. expected waiting time. 0.6. 0.2. 0.1. 2. 4. Ch 6. engchi 8. i n U. 10. v. 12. 14. number of servers s. Figure 5.5: Expected waiting time given service and abandonment. (γ1 = γ2 = 0.5, λ1 = λ2 = s/2, µ = 1). 33.

(41) 0.7 EW. ,FCFS. EW 1,a ,FCFS. 0.6. EW. expected waiting time. 1,s. 1,s. ,LCFS. EW 1,a ,LCFS. 0.5. 0.4. 0.3. 0.2. 0.1. 2. 4. 立. 政治大 6. 8. 10. 12. 14. number of servers s. ‧. ‧ 國. 學. Figure 5.6: Expected waiting time given service and abandonment in Jouini and Roubos [7]. (γ1 = γ2 = 0.5, λ1 = λ2 = s/2, µ = 1). 0.45. y. sit. EW 1,a ,FCFS EW 1,s ,LCFS. al. er. EW 1,a ,LCFS. n. 0.3. io. expected waiting time. 0.35. EW 1,s ,FCFS. Nat. 0.4. 0.25 0.2. Ch. engchi. i n U. v. 0.15 0.1 0.05 2. 4. 6. 8. 10. 12. 14. number of servers s. Figure 5.7: Expected waiting time given service and abandonment. (γ1 = γ2 = 1, λ1 = λ2 = s/2, µ = 1). 34.

(42) 0.45. EW. ,FCFS. EW 1,a ,FCFS. 0.4. EW. expected waiting time. 1,s. 0.35. 1,s. ,LCFS. EW 1,a ,LCFS. 0.3 0.25 0.2 0.15 0.1 0.05 2. 4. 政治大 6. 立. 8. 10. 12. 14. number of servers s. ‧. ‧ 國. 學. Figure 5.8: Expected waiting time given service and abandonment in Jouini and Roubos [7]. (γ1 = γ2 = 1, λ1 = λ2 = s/2, µ = 1). 0.07. y. sit. 0.05. EW 1,a ,FCFS EW 1,s ,LCFS EW 1,a ,LCFS. er. io. al. n. expected waiting time. Nat. 0.06. EW 1,s ,FCFS. 0.04. Ch. 0.03. engchi. i n U. v. 0.02. 0.01. 0 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. number of servers s. Figure 5.9: Expected waiting time given service and abandonment. (K = 15, ϵ = 10−10 , µ = 5, γ1 = 3, γ2 = 5, λ1 = 10, λ2 = 2s). 35.

(43) 0.07 EW. 0.06. ,FCFS. EW 1,a ,FCFS EW. expected waiting time. 1,s. 1,s. ,LCFS. EW 1,a ,LCFS. 0.05. 0.04. 0.03. 0.02. 0.01. 政治大. 0 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. number of servers s. 立. ‧ 國. ‧. EW 1,s ,FCFS. y. Nat. 400. 學. Figure 5.10: Expected waiting time given service and abandonment. (K = 15, ϵ = 10−10 , µ = 5, γ1 = 3, γ2 = 5, λ1 = 10, λ2 = 4s). sit. EW 1,a ,FCFS. io. al. er. EW 1,s ,LCFS. 350. time (sec). n. EW 1,a ,LCFS. 300. Ch. engchi. i n U. v. 250. 200. 150. 6. 7. 8. 9. 10. 11. 12. 13. 14. s. Figure 5.11: CPU time to compute the conditional expected waiting time in Figure 5.10. (K = 15, ϵ = 10−10 , µ = 5, γ1 = 3, γ2 = 5, λ1 = 10, λ2 = 4s). 36.

(44) Chapter 6 Conclusion 政治大 This thesis considers a system of multi-server queues with two classes of impatient customer: 立. ‧ 國. 學. high-priority and low-priority. Customers join the system according to a Poisson process and customers may abandon service after entering the queue for an exponentially distributed dura-. ‧. tion with distinct rates. First, we developed a method by which to compute the probability of all servers being idle. Next, we continue to compute the stationary probability of number of cus-. y. Nat. io. sit. tomers in both classes in order to derive the expected waiting time for both classes of customers.. n. al. er. For high-priority customers, we developed performance measures related to queueing times and. i n U. v. conditional waiting times in cases where service is provided and cases of abandonment. Future. Ch. engchi. research will be aimed at determining conditional waiting times for low-priority customers in cases where service is provided and in cases of abandonment. Researchers could also examine this model in cases where the two classes of customer differ with regard to service rates.. 37.

(45) Appendix A The probability of all server idles and LST 政治 of the virtual waiting time大立. ‧ 國. 學. In this Appendix, we will show that the different method to compute the probability of all. ‧. server idles given in section 5.1. In the Sarhangian and Balciog̃lu [14], it is given. y. j. ∑ ∏ λ2 g˜0 (kγ2 ))−1.. j =0 k =0. n. al. ∞. sit. io. i =0. λi λ c −1 λ + i!µi λ2 (c − 1)!µc−1. er. Nat. s −1. P0 = ( ∑. i n U. v. First, we defined f i ( x ) be the density function of virtual waiting time for class i. We have the following normalizing equation. Ch. engchi ∫ ∞. P (W = 0 ) +. 0. f i (y)dy = 1, i = 1, 2.. Next, employing the level-crossing theorem , we have the following equation f 1 ( x ) = λps−1 e. −sµx. + λ1. ∫ x 0. f 2 ( x ) = λps−1 B¯ 0,λ1 ( x ) + λ2. e. −sµ( x −y) −γ1 y. ∫ x 0. e. f 1 (y)dy + λ2 e. −sµx. B¯ 0,λ1 ( x − y)e−γ2 y f 2 (y)dy.. 38. ∫ ∞ 0. e−γ2 y f 1 (y)dy,.

(46) Taking the LST of its, we get sµ + t ˜ λ λ2 ˜ f˜1 (t + γ1 ) − f 1 ( t ) = − p s −1 − f 2 (γ2 ), λ1 λ1 λ1 f˜2 (t) λ f˜2 (t + γ2 ) − = − p s −1 , λ2 g˜0 (t) λ2 where g˜0 ( x ) is the LST of B¯ 0,λ1 ,γ1 with the complementary distribution function of an 0-busy period.. Applying Jagerman [6] , we have g˜ (t + kγ ). 政 ∑治∏ λ 大. λps−1 f˜2 (t) = λ2. 立. ∞. j. 2 0. 2. (A.0.1). j =0 k =0. E[ L ] =. Nat. K. λ1k. k =0. ∏kj=0 (sµ + jγ1 ). ∑. .. sit. c. y. Balciog̃lu [14], we have. ‧. ‧ 國. 學. Let E[ Lc ] denote the expected length of a busy period. From Eq.(22) in Sarhangian and. er. io. Also, we let s → 0 in Eq. (A.0.1), we have. al. n. v c i n C(1 − P(W = 0)) − λp s −1 E [ L ] U f˜2 (γ2 ) = h e n . i c h gλc2 E[ L ]. Because of s −1. P (W = 0 ) =. ∑. j =0. λj p0 , j!µ j. and ∫ ∞ 0. f 2 (y)dy = f˜2 (0),. 39. (A.0.2).

(47) we have s −1. P (W = 0 ) =. ∑. j =0. λj p0 = 1 − f˜2 (0), j!µ j. and s −1. P0 = ( ∑. i =0. λ λ c −1 λi + i!µi λ2 (c − 1)!µc−1. 立. ∞. j. ∑ ∏ λ2 g˜0 (kγ2 ))−1.. j =0 k =0. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 40. i n U. v.

(48) Appendix B Computation of R(n) 政治大 In the Phung-Duc and Kawanishi [11], it is given 立 ‧. ‧ 國. 學. Algorithm 1 Calculate R(n) Require: { An , Bn , Cn , k, ε} Ensure: R(n) k⇐1 (n) (n) Compute Gk and Gk+1 using Proposition 2.. ( n +1) (n) while Gk − Gk > ε do ∞ k ⇐ k+1 end while (n) R(n) ⇐ Gk. n. engchi. 41. sit er. io. Ch. y. Nat. al. i n U. v.

(49) Bibliography [1] F. Baccelli and G. Hebuterne. On queues with impatient customers. 1981. [2] A. Brandt and M. Brandt. On the m (n)/m (n)/s queue with impatient calls. Performance Evaluation, 35(1):1–18, 1999.. 立. 政治大. [3] B. D. Choi, B. Kim, and J. Chung. M/m/1 queue with impatient customers of higher. ‧ 國. 學. priority. Queueing Systems, 38(1):49–66, 2001.. ‧. [4] O. Garnett, A. Mandelbaum, and M. Reiman. Designing a call center with impatient cus-. sit. y. Nat. tomers. Manufacturing & Service Operations Management, 4(3):208–227, 2002.. al. er. io. [5] F. Iravani and B. Balcıog̃lu. On priority queues with impatient customers. Queueing. n. Systems, 58(4):239–260, 2008.. Ch. engchi. i n U. v. [6] D. L. Jagerman. Difference equations with applications to queues. CRC Press, 2000. [7] O. Jouini and A. Roubos. On multiple priority multi-server queues with impatience. Journal of the Operational Research Society, 65(5):616–632, 2013. [8] O. Jouini, Z. Aksin, and Y. Dallery. Call centers with delay information: Models and insights. Manufacturing & Service Operations Management, 13(4):534–548, 2011. [9] E. P. Kao and S. D. Wilson. Analysis of nonpreemptive priority queues with multiple servers and two priority classes. European Journal of Operational Research, 118(1):181– 193, 1999. [10] O. Kella and U. Yechiali. Waiting times in the non-preemptive priority m/m/c queue. Stochastic Models, 1(2):257–262, 1985. 42.

(50) [11] T. Phung-Duc and K. Kawanishi. Multiserver retrial queues with after-call work. Numerical Algebra, Control and Optimization, 1(4):639–656, 2011. [12] T. Phung-Duc, H. Masuyama, S. Kasahara, and Y. Takahashi. A simple algorithm for the rate matrices of level-dependent qbd processes. In Proceedings of the 5th international conference on queueing theory and network applications, pages 46–52. ACM, 2010. [13] T. Phung-Duc, H. Masuyama, S. Kasahara, and Y. Takahashi. A matrix continued fraction approach to multiserver retrial queues. Annals of Operations Research, 202(1):161–183, 2013.. 政治大. [14] V. Sarhangian and B. Balciog̃lu. Waiting time analysis of multi-class queues with impatient. 立. customers. Probability in the Engineering and Informational Sciences, 27(03):333–352,. ‧ 國. 學. 2013.. dom, 2003.. ‧. [15] A. Sleptchenko. Multi-class, multi-server queues with non-preemptive priorities. Euran-. sit. y. Nat. io. ational Research, 155(2):502–515, 2004.. n. al. Ch. engchi. er. [16] Q. Wang. Modeling and analysis of high risk patient queues. European Journal of Oper-. i n U. v. [17] S. Zeltyn, Z. Feldman, and S. Wasserkrug. Waiting and sojourn times in a multi-server queue with mixed priorities. Queueing Systems, 61(4):305–328, 2009.. 43.

(51)