Performance Analysis of a Telephone System with both Patient and Impatient Customers


 Clarissa Price
 1 years ago
 Views:
Transcription
1 Performance Analysis of a Telephone System with both Patient and Impatient Customers Yiqiang Quennel Zhao Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9 Attahiru Sule Alfa Department of Mechanical and Industrial Engineering University of Manitoba Winnipeg, Manitoba Canada R3T 2N2 September 2, 2004 Abstract: In this paper, the performance of a telephone system in which there are both patient and impatient customers is studied. Depending on the waiting time in the buffer, a customer may become impatient and therefore the call is incomplete. A impatient customer still consumes real processing time. The impact of impatient customers on the system is much more remarkable than people would expect. We formulate this model as a queueing system with a finite buffer. The status of a customer, i.e. whether patient or impatient, depends on the waiting time in the buffer and the service time is different for different type of customers. The joint distribution of the numbers of patient and impatient customers in the system is obtained for the system in equilibrium. Expressions of many performance measures, including the average number of patient customers or impatient customers in the system, the average length of the service time, the average arrivals during a service period and the proportion of the useful service time contributing towards patient customers service, can be expressed in terms of the joint probabilities. The waiting time probabilities and the average waiting time of a customer in the buffer, and the probability 1
2 that a customer will be served as a patient customer are also obtained. Numerical examples are presented. 1 Introduction The traffic analysis of telephone calls to a local electronic switching system is essential for the telephone company to provide customers with high quality of services. Especially, the system should maintain a high level of successfully switched calls under the heavy, or even overloaded, traffic. An incomplete call (impatient customer), either due to a dial tone delay or abandonment from the system, still consumes about thirty to eighty percent of the real time of processing a complete call. This is a very remarkable impact that the impatient customers has on the system. Study of field data [4] showed that only about 20% of customers are patient after having waited for 12 seconds. The traditional first in first out (FIFO) strategy will cause a severe throughput degradation under the heavy traffic. Early simulation results on the telephone systems [1] showed that a last in first out (LIFO) queue discipline will keep the successfully carried load close to system capacity under service overload. Besides the study of field data and the simulation results, numerical results for some LIFO systems were obtained in terms of analysing continuous time queueing models [2]. In this paper, we propose a discrete time queueing model to analyze the performance of the telephone system with a LIFOpushout queue discipline. We study a single local electronic switching system with a finite buffer. Customers arrive randomly and all of them are initially patient. Whether or not a customer will become impatient depends on the waiting time it has endured in the buffer. The service time is different for different type of customers; it is longer for a patient customer. Because of the LIFOpushout queue discipline, a customer, which could not get a service upon its arrival, will enter the system and wait in the buffer for the service no matter whether or not the buffer is full. When the buffer is full, the customer(s), who came earliest, will be removed from the system. An approximate mathematical model is formulated and the system equations are provided for computing the joint distribution of the numbers of patient and impatient customers in the system for the system in equilibrium in Section 2. Expressions for computing the average number of patient customers and impatient customers in the system, the average length 2
3 of the service time, the average arrivals during a service period and the proportion of the service time contributing to patient customers are derived in terms of the equilibrium probabilities in Section 3. The waiting time probabilities and the average waiting time of a customer in the buffer, and the probability that a customer will be served as a patient customer are also studied in Section 3. Numerical results of performance measures and the performance analysis based on the numerical results of the system are provided in Section 4. 2 Mathematical Model We now study the telephone system introduced in Section 1 by making the following assumptions. Customers (or calls) arrive according to a Poisson process, or the interarrival times are independently identically exponentially distributed random variables. There is only one server (telephone switch facility) in the system and the service time of a customer depends on its waiting time already endured in the system. This assumption is based on the fact that a customer may become impatient after having waited for a certain amount of time for a dial tone and may either dial the number before hearing a dial tone or abandon the system. We call them impatient customers. An impatient customer still consumes about 30 to 80% of the real time of processing a patient customer [2]. The impact of the impatient customers would become very much more significant as the traffic becomes heavier. There is a buffer of finite size in the system. A customer who could not get into the service upon the arrival will be waiting in the buffer. Because a patient customer may become impatient later, the number of patient customers or the number of impatient customers in the system will not be a Markov process since we have to trace the waiting time a patient customer has endured. We propose an approximate model to perform the analysis of the impact of the impatient customers on the system. Consider the imbedded times at the service completion. Let t n be the time of the nth service completion. The arrivals: We assume that during a service time period all customers arrive at the same moment immediately after the time epoch t n. The number of customers arriving during the service time of the nth customer is a Poisson distributed random variable with the parameter 3
4 λ(t n t n 1 ). Customers will be served according to the nonpreemptive last come first served (LIFO) discipline. A customer will become impatient after having waited for T 0 (> 0) time units in the buffer. The justification of the above assumptions is based on the following facts. The service time is relatively much smaller than the waiting time threshold T 0 (T 0 is ranging from a few seconds to teenseconds according to field data). There is little real need to distinguish the arrival moments during the service period. And the LIFO queue discipline is believed to be the better strategy than the FIFO discipline, since the freshest call is the most likely one to finish the call. The service: The service time T of a customer depends on the waiting time W it has endured in the system. Specifically, we assume that T, if W < T 0 T = T +, it W T 0, where T 0 is the waiting time threshold defined earlier, and T and T + are two positive numbers. Since an impatient customer would consume about 30 to 80% of the real time of processing a patient customer and the service time is much less than the waiting time threshold, T + < T << T 0. For convenience, we further assume that T 0 /T is an integer. The waiting time: If a customer has waited in the buffer for more than T 0 units of time, he is an impatient customer and will remain impatient. But a customer who has only waited for less than T 0 units of time may become impatient later. Therefore, we need to trace the waiting time in the buffer of every patient customer, which would lead to a very complex model. Instead of doing it, we refresh all patient customers at each service completion epoch by forgetting their waiting time history. In other words, the waiting time a patient customer has endured will be reset to zero at the next service completion time. This assumption can be justified under heavy traffic conditions as follows. Let p 0 be the probability that there is no customer arriving during the service time T: p 0 = E(e λt ). Under heavy traffic conditions or in overloaded cases, p 0 is small and therefore, most likely, at least one customer would arrive. This means that, since a LIFO discipline is used, most of 4
5 patient customers left behind at the service completion will finally become impatient since they cannot be served by time T 0 no matter what their waiting time history is. Since the behaviour of the system under heavy and overloaded traffic conditions is our main concern in this paper, our assumption can be justified. The buffer: The maximum number of patient customers in the system is K = T 0 /T. Let K + be the maximum number of impatient customers in the system. The exact value of K + is the buffer size minus the number of patient customers in the system. Since the buffer size is usually large, several hundred and more (see [2]), and K is relatively small, we can simply assume that the maximum number of impatient customers in the system is independent of the number of patient customers in the system. As soon as the number of the impatient customers becomes larger than K +, the earliest arrival(s) will be removed from the buffer. If there are more than K + customers arrived in the same batch, the customer(s), which will be removed from the system, will be randomly selected. Let N + (n) and N (n) be, respectively, the numbers of impatient and patient customers in the system at time t n. Let A n+1 be the number of customers arriving during the time period from t n to t n+1, which depends on the service time. And let (a, b) + and (a, b) be, respectively, max(a, b) and min(a, b). Then we have ((0, N + (n) 1) + + (0, A n+1 K ) +, K + ), if N (n) = 0 N + (n + 1) = (N + + (0, N 1 + A n+1 K ) +, K + ), if N > 0 (1) and (A n+1, K ), if N (n) = 0 N (n + 1) = (N (n) 1 + A n+1, K ), if N > 0. (2) Therefore, {(N + (n), N (n));n = 0, 1, 2,.....} is a Markov chain with the state space S = {(i, j) i = 0, 1,...,K + and j = 0, 1,...,K }. The transition matrix P = (p (i,j) (s,t) ) can be explicitly found by using the relationships given in (1) and (2) and by noticing that the service time T = T if a) N (n) = N + (n) = 0 or b) N (n) > 0; and T = T + if N + (n) > N (n) = 0. The important measures of the system performance include the average number of patient customers in the system, the average number of impatient customers in the system, 5
6 the average length of the service time, the average arrivals during a service period, the proportion of the service time contributing to patient customers, the average waiting time of a customer before entering the service and so forth. The study of all other performance measures except the average waiting time will be carried out in terms of the joint equilibrium probabilities of the numbers of two type customers in the system, which will be obtained by solving the following system stationary equations. The average waiting time will be treated separately. The Stationary equations: Let p i,j = lim n P {N + (n) = i, N (n) = j} be the joint equilibrium probability that there are i and j, respectively, impatient and patient customers in the system. For k = 0, 1, 2,..., define a k = (λt ) k e λt k! and b k = (λt +) k e λt +. (3) k! Then the stationary equations of the system are written as p 0,j = a j p 0,0 + p i,j = p K+,j = j a j k p 0,k+1 + b j p 1,0, j = 0, 1,...,K 1, k=0 j a j k p i,k+1 + b j p i+1,0, i = 1, 2,...,K + 1, j = 0, 1,...,K 1, k=0 j k=0 p i,k = a K +i p 0,0 + a j k p K+,k+1, j = 0, 1,...,K 1 (4) i l=0 p K+,K = α K +K + p 0,0 + K 1 K + l=0 k=0 K 1 k=0 a (i l)+(k k) p l,k+1 + i b (i l)+k p l+1,0, l=0 i = 0, 1,...,K + 1, K + 1 α (K+ l)+(k k) p l,k+1 + l=0 β (K+ l)+k p l+1,0, where α n = a k and β n = b k for n = 0, 1, 2,.... (5) k=n k=n 3 Performance measures A number of interesting performance measures are studied in this section, including the average number of patient customers or impatient customers in the system, the average 6
7 length of the service time, the average arrivals during a service period, the proportion of the service time contributing to patient customers, the waiting time probabilities and the average waiting time of a customer in the buffer before entering the service, the probability that a customer will be served as a patient customer, and so forth. The computation of the equilibrium probabilities p i,j will be treated in Section 4. The numbers of patient and impatient customers in the system: The probability distributions of the numbers of patient and impatient customers in the system are, respectively, computed according to the following expressions. K + p j = P {N = j} = p i,j (6) and K p + i = P {N + = i} = p i,j, (7) where N and N + are the equilibrium numbers of patient and impatient customers in the system respectively. The average numbers of the patient and impatient customers in the system are, respectively, computed by K i=0 j=0 E(N ) = jp j (8) j=1 and K + E(N + ) = ip + i. (9) i=1 Proportions of the service time spent on processing patient and impatient customers: Since an impatient customer still consumes about 30 to 80 percent of real time of processing a patient customer, it is important to know the proportion of service time spent on processing a patient or an impatient customer. Upon the completion of a service, the next service time will be spent on processing a patient customer whenever there is at least one patient customer in the system or the system is empty. The proportion p w of the useful service time contributing towards patient customers service is defined as the probability that there is at least one patient customer in the system or the system is 7
8 empty. Or p w = 1 p i,0 = 1 p 0 + p 0,0. i=1 q w = 1 p w = p 0 p 0,0 is the proportion of the service time contributing to impatient customers. The number of customers arriving during the service time: The number A of customers arriving during the service time depends on the length of the service time. By conditioning on the length of the service time, we can find (λt ) k P {A = k} = p w e λt (λt + ) k + q w e λt +. (10) k! k! The average number E(A) of customers arriving during a service time is given by E(A) = λe(t) = λ(p w T + q w T + ). The average waiting time in the buffer: The waiting time of a customer in the buffer is a very important performance measure of the system. Instead of using the average number of customers for the approximate model to estimate the average waiting time, we use a different method here. We use this different method to study the waiting time in the buffer of a randomly selected (called tagged) customer in an arrival batch, which will be served as a patient customer. Let A be the batch size the tagged customer belongs to, then P {A = j} = jp {A = j A > 0} jp {A = j} i=1 =. (11) ip {A = i A > 0} E(A) Let N < +1 be the place of the tagged customer in the arrival batch. Obviously, if N < = 0 the tagged customer will be the customer first served in the batch. Conditioning on the number A of customers in the tagged batch, we have P {N < = k} = j=k+1 P {A = j} j = P {A k + 1} E(A), for k = 0, 1, 2,.... (12) Since T + = (%30 to %80) T << T 0, we can simply use the average service time E(T) for both T and T +. Numerical results showed us that for different values of λ (from light traffic to heavy traffic and to overloaded), the average service time is close to T. It 8
9 means that a customer will become impatient after having waited T 0 K E(T). Let W be the waiting time of the tagged customer in the buffer before entering the service, and let P {W = n} be the probability that the waiting time is equal to ne(t). Then P {W = 0} = P {N < = 0} = For n = 1, 2,...,K 1, conditioning on N <, we have P {W = n} = P {A 1}. (13) E(A) n P {N < = k}p {W = n N < = k}. (14) k=1 The conditional probability P {W = n N < = k} can be determined as follows. Let t be the current time, then n P {W = n N < = k} = P A t+i n k = 0, i=1 j A t+i j k > 0 for j = 1, 2,...,n 1. i=1 (15) Conditioning on the numbers of customers arriving in the following service periods and noticing that arrivals during different service periods are independent, P {W = n N < = k} = P {A = 0} n k k 1 =(2 k) + n k k 1 k 2 =(3 k k 1 ) + n k k 1 k n 2 k n 1 =(n k k 1 k n 2 ) + P {A = k 1 }P {A = k 2 } P {A = k n 1 }. (16) Using probabilities P {W = n}, we can determine the probability P 1 that the tagged customer will be served as a patient customer: P 1 = K 1 n=0 P {W = n} (17) and the probability P 1 that the tagged customer will be served as a patient customer but not the first customer being served in the tagged batch: P 1 = K 1 n=1 P {W = n}. (18) The average waiting time of the tagged customer in the buffer before entering the service given that it will be finally served as a patient customer is given by E(W W < K ) = E(T) P 1 9 K 1 n=1 np {W = n}. (19)
10 And the average waiting time of the tagged customer in the buffer before entering the service given that it will be finally served as a patient customer but not the first customer being served in the tagged batch is given by E(W 0 < W < K ) = E(T) P 1 K 1 n=1 np {W = n}. (20) Some other important performance measures can also be found similarly. For example, by using a similar conditional probability argument as in obtaining the waiting time, we can give an expression for the probability P 2 that the tagged customer, which is not entering the service upon its arrival, will be finally removed from the system due to the buffer becoming full. Specifically, we condition on N < and then condition on the epochs at which the tagged one will be removed from the system. Therefore, we can find the probability P 3 that the tagged customer will be served as an impatient customer: P 3 = 1 P 1 P 2. 4 Numerical results The model formulated in the previous section enable us to use different numerical procedures to computing the joint equilibrium probabilities the number of two type customers in the system. We used the statereduction method [3], which is numerically stable. Let us rewrite the stationary equations given in Section 2 in the matrix form, which is more explanatory. Let p = ( p 0, p 1,..., p K+ ) with p i = (p i0, p i1,...,p i K ) for i = 0, 1,...,K +, then pp = p with p e = 1, where e is the transpose of the row vector e of size (K + +1) (K 1) with all components equal to one. P is the transition matrix given by B 0 B 1 B 2 B 3 B K+ 1 B K + A 0 A 1 A 2 A 3 A K+ 1 A K + 0 A 0 A 1 A 2 A K+ 2 A K P = A 0 A 1 A K+ 3 A, (21) K A 0 A 1 10
11 where a 0 a 1 a 2 a K 1 a K a 0 a 1 a 2 a K 1 a K 0 a 0 a 1 a K 2 a K 1 B 0 =, (22) 0 0 a 0 a K 3 a K a 0 a a j+k a j+k a j+k 1 B j = (23) a j+k a j+1 for j = 1, 2,...,K + 1, α K+ +K α K+ +K B K α K+ +K 1 + =, (24) α K+ +K α K+ +1 b 0 b 1 b 2 b K 1 b K A 0 =, (25) b K +1 a 0 a 1 a 2 a K 1 a K A 1 = 0 a 0 a 1 a K 2 a K 1, (26) a 0 a 1 11
12 0 0 0 b j+k a j+k 1 A j = a j+k a j for j = 2, 3,...,K + 1, A 1 = β K +1 a 0 a 1 a 2 a K 1 α K 0 a 0 a 1 a K 2 α K a 0 α 1 (27), (28) and A j = β j+k α j+k α j+k (29) α j for j = 2, 3,...,K +. This is a matrix of the truncated M/G/1 type, which is a special case discussed in [3]. We consider the following numerical example. Values of different parameters are now determined. First of all, for convenience, let T + = 1 time unit. Since the service time T + spent on an impatient customer is about 30 to 80 percent of the real time of processing a patient customer, let T = 1.5 time units. According to the information of the field study that either 0% dial tone delays greater than 3 seconds or nearly 100% and that most of customers would become impatient after having waited for more than 4 seconds (see [2]), we let 4 seconds be the threshold value T 0 of the waiting time a customer can endure, or approximately T 0 = 15 time units here. K = T 0 /T = 10. And finally let K + = 100. For different values of the arrival rate λ, interesting performance measures are computed by using the results given in previous sections. If λe(t) = 1, then the average number of the patient customers arriving during the service time is 1, which means that the system is approximately saturated. The system 12
13 is overloaded if λe(t) > 1. In Figure 1, the average numbers of patient and impatient customers vs the arrival rate are illustrated. When the value of λ increases to about 0.68, the system is saturated. The average number of impatient customers in the system is almost 100 (full). In contrast with it, the average number of patient customers in the system is only about 5.5. Figure 2 provides the probability of the system being empty vs the arrival rate. As we expected, this probability is almost zero when the system is saturated. The proportion of the service time contributing to patient customers vs the arrival rate is shown in Figure 3. At λ = 0.4, the probability of the system being empty is about 0.40 (see Figure 2), which means that 60% of time, the server is idle. But, when the server is busy it will serve a patient customer with probability almost equal to 1. Under very heavy traffic or overloaded conditions, the probability that the server will serve a patient customer is also very large (almost 1 if the value of λ excesses 0.80 or ρ > 1.20). In this case, the server is almost always busy (p 0,0 < ). This result showed us that the LIFO strategy keeps the successfully switched calls close to system capacity. Even in the worst case the probability that the server will serve a patient customer is still larger than The probability that an arrival will be finally served as a patient customer is almost 1 at λ = 0.1 and decreases as λ increases, which is showed by Figure 4. The probability that an arrival will be finally served as a patient customer but not the first one served in the batch vs λ is given in Figure 5. This probability increase first as λ increases until about at λ = 0.90 and then decreases. In both light and heavy traffic, if an arrival cannot be served upon the arrival, with a small probability that it can be served as a patient customer later. Finally, in Figure 6, vs the arrival rate is provided the average waiting time of an arrival, which will be finally served as a patient customer but not the first one served in the batch. It is the product of two factors: E(T) and W f = 1 P 1 K 1 n=1 np {W = n}. As λ increases, E(T) decreases first and then increases, whereas W f is in the opposite way. The change of the waiting time as the arrival rate is dominated by the factor W f for almost all values of λ, except the values around the system saturation point (λ 0.68 or ρ 1.0), at which E(T) takes its minimum. This resulted in two peaks in the graph. 13
14 Acknowledgement The research of Y.Q. Zhao is supported in part by NSERC grant No. 4452, and that of A.S. Alfa by NSERC grant No. OGP and a grant from BellNorthern Research. The authors acknowledge Dr. W.K. Grassmann for providing them with programs used for computing the equilibrium probabilities based on the statereduction method, and thank the referee for valuable comments. References [1] L. Burkard, J.J. Phelan and M.D. Weekly, Customer behavior and unexpected dial tone delay, Proc. 10th ITC, Montreal, 1983, paper No.5. [2] L.J. Forys, Performance analysis of a new overload strategy, Proc. 10th ITC, Montreal, 1983, paper No.4. [3] W.K. Grassmann and D.P. Heyman, Computation of steadystate probabilities for infinitestate Markov chains with repeating rows, ORSA J. on Computing, Vol. 5, No. 3 (1993), pp [4] R.I. Wilkinson, Theories for toll traffic engineering in the U.S.A., Bell System Technical Journal, March (1956), pp
15 Figure 1: The average numbers of patient and impatient customers in the system vs the arrival rate. 15
16 Figure 2: The probability of the system being empty vs the arrival rate. 16
17 Figure 3: The proportion of the service time contributing to patient customers vs the arrival rate. 17
18 Figure 4: The probability that an arrival will be finally served as a patient customer vs the arrival rate. 18
19 Figure 5: The probability that an arrival will be finally served as a patient customer but not the first one served in the arrival batch vs the arrival rate. 19
20 Figure 6: The average waiting time of an arrival, which will be finally served as a patient customer but not the first one served in the batch vs the arrival rate. 20
4 The M/M/1 queue. 4.1 Timedependent behaviour
4 The M/M/1 queue In this chapter we will analyze the model with exponential interarrival times with mean 1/λ, exponential service times with mean 1/µ and a single server. Customers are served in order
More informationUNIT 2 QUEUING THEORY
UNIT 2 QUEUING THEORY LESSON 24 Learning Objective: Apply formulae to find solution that will predict the behaviour of the single server model II. Apply formulae to find solution that will predict the
More informationModelling the performance of computer mirroring with difference queues
Modelling the performance of computer mirroring with difference queues Przemyslaw Pochec Faculty of Computer Science University of New Brunswick, Fredericton, Canada E3A 5A3 email pochec@unb.ca ABSTRACT
More informationM/M/1 and M/M/m Queueing Systems
M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General  can be any distribution. First letter: Arrival process; M: memoryless  exponential
More informationThe Exponential Distribution
21 The Exponential Distribution From DiscreteTime to ContinuousTime: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More informationThe Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback
The Joint Distribution of Server State and Queue Length of M/M/1/1 Retrial Queue with Abandonment and Feedback Hamada Alshaer Université Pierre et Marie Curie  Lip 6 7515 Paris, France Hamada.alshaer@lip6.fr
More informationPull versus Push Mechanism in Large Distributed Networks: Closed Form Results
Pull versus Push Mechanism in Large Distributed Networks: Closed Form Results Wouter Minnebo, Benny Van Houdt Dept. Mathematics and Computer Science University of Antwerp  iminds Antwerp, Belgium Wouter
More informationOptimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani. EPEW 2014, Florence
Optimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani EPEW 2014, Florence Scenario How many cloud instances should be hired? Requests Host hiring servers The number of active servers is controlled
More informationIEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS
IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)
More information6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory
6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory Massachusetts Institute of Technology Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS
More informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More informationDPolicy for a Production Inventory System with Perishable Items
Chapter 6 DPolicy for a Production Inventory System with Perishable Items 6.1 Introduction So far we were concentrating on invenory with positive (random) service time. In this chapter we concentrate
More informationQueuing Model Dr. Yifeng Zhu. In queueing theory, the average number of tasks in a stable system (over some time interval), N, is given by
Note 3: M/M/ April, 007 ECE598 Advanced Computer Architecture URL: http://www.eece.maine.edu/ zhu/ece598/ Queuing Model Dr. Yifeng Zhu Little s law In queueing theory, the average number of tasks in a
More informationPerformance Analysis of Computer Systems
Performance Analysis of Computer Systems Introduction to Queuing Theory Holger Brunst (holger.brunst@tudresden.de) Matthias S. Mueller (matthias.mueller@tudresden.de) Summary of Previous Lecture Simulation
More informationDiscreteEvent Simulation
DiscreteEvent Simulation Prateek Sharma Abstract: Simulation can be regarded as the emulation of the behavior of a realworld system over an interval of time. The process of simulation relies upon the
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More information1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easytocheck
More informationQUEUING THEORY. 1. Introduction
QUEUING THEORY RYAN BERRY Abstract. This paper defines the building blocks of and derives basic queuing systems. It begins with a review of some probability theory and then defines processes used to analyze
More informationContinuousTime Markov Chains  Introduction
25 ContinuousTime Markov Chains  Introduction Prior to introducing continuoustime Markov chains today, let us start off with an example involving the Poisson process. Our particular focus in this example
More informationBonusmalus systems and Markov chains
Bonusmalus systems and Markov chains Dutch car insurance bonusmalus system class % increase new class after # claims 0 1 2 >3 14 30 14 9 5 1 13 32.5 14 8 4 1 12 35 13 8 4 1 11 37.5 12 7 3 1 10 40 11
More informationAn Introduction to Queueing Theory
An Introduction to Queueing Theory Rein Nobel Department of Econometrics, Vrije Universiteit, Amsterdam Open Middag november 20 Overview. Basic results for queueing models in continuous time: (a) delay
More informationStochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations
56 Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations FlorinCătălin ENACHE
More informationChapter 2. Simulation Examples 2.1. Prof. Dr. Mesut Güneş Ch. 2 Simulation Examples
Chapter 2 Simulation Examples 2.1 Contents Simulation using Tables Simulation of Queueing Systems Examples A Grocery Call Center Inventory System Appendix: Random Digits 1.2 Simulation using Tables 1.3
More informationMarkov Chains. Chapter Introduction and Definitions 110SOR201(2002)
page 5 SOR() Chapter Markov Chains. Introduction and Definitions Consider a sequence of consecutive times ( or trials or stages): n =,,,... Suppose that at each time a probabilistic experiment is performed,
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationAnalysis of a Production/Inventory System with Multiple Retailers
Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University
More informationHow Useful Is Old Information?
6 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 11, NO. 1, JANUARY 2000 How Useful Is Old Information? Michael Mitzenmacher AbstractÐWe consider the problem of load balancing in dynamic distributed
More informationStochastic Models for Inventory Management at Service Facilities
Stochastic Models for Inventory Management at Service Facilities O. Berman, E. Kim Presented by F. Zoghalchi University of Toronto Rotman School of Management Dec, 2012 Agenda 1 Problem description Deterministic
More informationUser s Guide for ContactCenters Simulation Library
User s Guide for ContactCenters Simulation Library Generic Simulator for Blend and Multiskill Call Centers Version: March 17, 2014 Eric Buist This document introduces a generic simulator for blend and
More informationOn Admission Control Policy for Multitasking Livechat Service Agents Researchinprogress Paper
On Admission Control Policy for Multitasking Livechat Service Agents Researchinprogress Paper Paulo Goes Dept. of Management Information Systems Eller College of Management, The University of Arizona,
More informationMarkov Chains and Queueing Networks
CS 797 Independent Study Report on Markov Chains and Queueing Networks By: Vibhu Saujanya Sharma (Roll No. Y211165, CSE, IIT Kanpur) Under the supervision of: Prof. S. K. Iyer (Dept. Of Mathematics, IIT
More informationContinuoustime Markov Chains
Continuoustime Markov Chains Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 31, 2016
More informationA QUEUEINGINVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND. bhaji@usc.edu
A QUEUEINGINVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND Rasoul Hai 1, Babak Hai 1 Industrial Engineering Department, Sharif University of Technology, +98166165708, hai@sharif.edu Industrial
More informationA Quantitative Approach to the Performance of Internet Telephony to Ebusiness Sites
A Quantitative Approach to the Performance of Internet Telephony to Ebusiness Sites Prathiusha Chinnusamy TransSolutions Fort Worth, TX 76155, USA Natarajan Gautam Harold and Inge Marcus Department of
More informationOPTIMIZED PERFORMANCE EVALUATIONS OF CLOUD COMPUTING SERVERS
OPTIMIZED PERFORMANCE EVALUATIONS OF CLOUD COMPUTING SERVERS K. Sarathkumar Computer Science Department, Saveetha School of Engineering Saveetha University, Chennai Abstract: The Cloud computing is one
More informationWhen Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance
When Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance Mor Armony 1 Itay Gurvich 2 July 27, 2006 Abstract We study crossselling operations in call centers. The following
More informationSTABILITY OF LUKUMAR NETWORKS UNDER LONGESTQUEUE AND LONGESTDOMINATINGQUEUE SCHEDULING
Applied Probability Trust (28 December 2012) STABILITY OF LUKUMAR NETWORKS UNDER LONGESTQUEUE AND LONGESTDOMINATINGQUEUE SCHEDULING RAMTIN PEDARSANI and JEAN WALRAND, University of California, Berkeley
More informationCHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction
CHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction Consider in this chapter a business organization under fluctuating conditions of availability of manpower and business
More informationStructure Preserving Model Reduction for Logistic Networks
Structure Preserving Model Reduction for Logistic Networks Fabian Wirth Institute of Mathematics University of Würzburg Workshop on Stochastic Models of Manufacturing Systems Einhoven, June 24 25, 2010.
More informationSingle item inventory control under periodic review and a minimum order quantity
Single item inventory control under periodic review and a minimum order quantity G. P. Kiesmüller, A.G. de Kok, S. Dabia Faculty of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513,
More informationConclusions and Suggestions for Future Research
6 Conclusions and Suggestions for Future Research In this thesis dynamic inbound contact centers with heterogeneous agents and retrials of impatient customers were analysed. The term dynamic characterises
More informationWhen Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance
When Promotions Meet Operations: CrossSelling and Its Effect on CallCenter Performance Mor Armony 1 Itay Gurvich 2 Submitted July 28, 2006; Revised August 31, 2007 Abstract We study crossselling operations
More informationCorrected Diffusion Approximations for the Maximum of HeavyTailed Random Walk
Corrected Diffusion Approximations for the Maximum of HeavyTailed Random Walk Jose Blanchet and Peter Glynn December, 2003. Let (X n : n 1) be a sequence of independent and identically distributed random
More informationA Markovian Sensibility Analysis for Parallel Processing Scheduling on GNU/Linux
A Markovian Sensibility Analysis for Parallel Processing Scheduling on GNU/Linux Regiane Y. Kawasaki 1, Luiz Affonso Guedes 2, Diego L. Cardoso 1, Carlos R. L. Francês 1, Glaucio H. S. Carvalho 1, Solon
More informationThe 8th International Conference on ebusiness (inceb2009) October 28th30th, 2009
ENHANCED OPERATIONAL PROCESS OF SECURE NETWORK MANAGEMENT SongKyoo Kim * Mobile Communication Divisions, Samsung Electronics, 94 ImsooDong, Gumi, Kyungpook 730350, South Korea amang.kim@samsung.com
More informationExponential Distribution
Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1
More informationINTEGRATED OPTIMIZATION OF SAFETY STOCK
INTEGRATED OPTIMIZATION OF SAFETY STOCK AND TRANSPORTATION CAPACITY Horst Tempelmeier Department of Production Management University of Cologne AlbertusMagnusPlatz D50932 Koeln, Germany http://www.spw.unikoeln.de/
More informationReinforcement Learning
Reinforcement Learning LU 2  Markov Decision Problems and Dynamic Programming Dr. Martin Lauer AG Maschinelles Lernen und Natürlichsprachliche Systeme AlbertLudwigsUniversität Freiburg martin.lauer@kit.edu
More informationTHE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:
THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces
More informationSimple Markovian Queueing Systems
Chapter 4 Simple Markovian Queueing Systems Poisson arrivals and exponential service make queueing models Markovian that are easy to analyze and get usable results. Historically, these are also the models
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
12th International Congress on Insurance: Mathematics and Economics July 1618, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint
More informationService Performance Analysis and Improvement for a Ticket Queue with Balking Customers. Long Gao. joint work with Jihong Ou and Susan Xu
Service Performance Analysis and Improvement for a Ticket Queue with Balking Customers joint work with Jihong Ou and Susan Xu THE PENNSYLVANIA STATE UNIVERSITY MSOM, Atlanta June 20, 2006 Outine Introduction
More informationHydrodynamic Limits of Randomized Load Balancing Networks
Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli
More informationRulebased Traffic Management for Inbound Call Centers
Vrije Universiteit Amsterdam Research Paper Business Analytics Rulebased Traffic Management for Inbound Call Centers Auteur: Tim Steinkuhler Supervisor: Prof. Dr. Ger Koole October 7, 2014 Contents Preface
More information1. Repetition probability theory and transforms
1. Repetition probability theory and transforms 1.1. A prisoner is kept in a cell with three doors. Through one of them he can get out of the prison. The other one leads to a tunnel: through this he is
More informationReinforcement Learning
Reinforcement Learning LU 2  Markov Decision Problems and Dynamic Programming Dr. Joschka Bödecker AG Maschinelles Lernen und Natürlichsprachliche Systeme AlbertLudwigsUniversität Freiburg jboedeck@informatik.unifreiburg.de
More informationAnalysis of Call Center Data
University of Pennsylvania ScholarlyCommons Wharton Research Scholars Journal Wharton School 412004 Analysis of Call Center Data Yu Chu Cheng University of Pennsylvania This paper is posted at ScholarlyCommons.
More informationRobust Staff Level Optimisation in Call Centres
Robust Staff Level Optimisation in Call Centres Sam Clarke Jesus College University of Oxford A thesis submitted for the degree of M.Sc. Mathematical Modelling and Scientific Computing Trinity 2007 Abstract
More informationQ UEUING ANALYSIS. William Stallings
Q UEUING ANALYSIS William Stallings WHY QUEUING ANALYSIS?...2 QUEUING MODELS...3 The SingleServer Queue...3 Queue Parameters...4 The Multiserver Queue...5 Basic Queuing Relationships...5 Assumptions...5
More informationMath 312 Lecture Notes Markov Chains
Math 312 Lecture Notes Markov Chains Warren Weckesser Department of Mathematics Colgate University Updated, 30 April 2005 Markov Chains A (finite) Markov chain is a process with a finite number of states
More informationWeb Hosting Service Level Agreements
Chapter 5 Web Hosting Service Level Agreements Alan King (Mentor) 1, Mehmet Begen, Monica Cojocaru 3, Ellen Fowler, Yashar Ganjali 4, Judy Lai 5, Taejin Lee 6, Carmeliza Navasca 7, Daniel Ryan Report prepared
More informationLECTURE 4. Last time: Lecture outline
LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random
More informationWeb Server Software Architectures
Web Server Software Architectures Author: Daniel A. Menascé Presenter: Noshaba Bakht Web Site performance and scalability 1.workload characteristics. 2.security mechanisms. 3. Web cluster architectures.
More informationQueueing Systems. Ivo Adan and Jacques Resing
Queueing Systems Ivo Adan and Jacques Resing Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 26, 2015 Contents
More informationNOVEL PRIORITISED EGPRS MEDIUM ACCESS REGIME FOR REDUCED FILE TRANSFER DELAY DURING CONGESTED PERIODS
NOVEL PRIORITISED EGPRS MEDIUM ACCESS REGIME FOR REDUCED FILE TRANSFER DELAY DURING CONGESTED PERIODS D. Todinca, P. Perry and J. Murphy Dublin City University, Ireland ABSTRACT The goal of this paper
More informationResearch Article Average Bandwidth Allocation Model of WFQ
Modelling and Simulation in Engineering Volume 2012, Article ID 301012, 7 pages doi:10.1155/2012/301012 Research Article Average Bandwidth Allocation Model of WFQ TomášBaloghandMartinMedvecký Institute
More informationCall Center  Supervisor Application User Manual
Forum 700 Call Center Supervisor Application User Manual Legal notice: Belgacom and the Belgacom logo are trademarks of Belgacom. All other trademarks are the property of their respective owners. The information
More informationSchool of Computer Science
DDSS:Dynamic Dedicated Servers Scheduling for Multi Priority Level Classes in Cloud Servers Husnu S. Narman, Md. Shohrab Hossain, Mohammed Atiquzzaman TROUTNRL13 Sep 13 Telecommunication & Network
More informationVoice Service Support over Cognitive Radio Networks
Voice Service Support over Cognitive Radio Networks Ping Wang, Dusit Niyato, and Hai Jiang Centre For Multimedia And Network Technology (CeMNeT), School of Computer Engineering, Nanyang Technological University,
More informationUser Manual. Call Center  Supervisor Application
User Manual Call Center  Supervisor Application Release 8.0  September 2010 Legal notice: Alcatel, Lucent, AlcatelLucent and the AlcatelLucent logo are trademarks of AlcatelLucent. All other trademarks
More informationCommon Approaches to RealTime Scheduling
Common Approaches to RealTime Scheduling Clockdriven timedriven schedulers Prioritydriven schedulers Examples of priority driven schedulers Effective timing constraints The EarliestDeadlineFirst
More informationLectures 5 & / Introduction to Queueing Theory
Lectures 5 & 6 6.263/16.37 Introduction to Queueing Theory MIT, LIDS Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS PS PS Packet Network PS PS
More informationExperiments on the local load balancing algorithms; part 1
Experiments on the local load balancing algorithms; part 1 Ştefan Măruşter Institute eaustria Timisoara West University of Timişoara, Romania maruster@info.uvt.ro Abstract. In this paper the influence
More informationWorked examples Random Processes
Worked examples Random Processes Example 1 Consider patients coming to a doctor s office at random points in time. Let X n denote the time (in hrs) that the n th patient has to wait before being admitted
More informationCALL CENTER PERFORMANCE EVALUATION USING QUEUEING NETWORK AND SIMULATION
CALL CENTER PERFORMANCE EVALUATION USING QUEUEING NETWORK AND SIMULATION MA 597 Assignment K.Anjaneyulu, Roll no: 06212303 1. Introduction A call center may be defined as a service unit where a group of
More informationNetwork Design Performance Evaluation, and Simulation #6
Network Design Performance Evaluation, and Simulation #6 1 Network Design Problem Goal Given QoS metric, e.g., Average delay Loss probability Characterization of the traffic, e.g., Average interarrival
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More informationRandom access protocols for channel access. Markov chains and their stability. Laurent Massoulié.
Random access protocols for channel access Markov chains and their stability laurent.massoulie@inria.fr Aloha: the first random access protocol for channel access [Abramson, Hawaii 70] Goal: allow machines
More informationQueuing Theory II 2006 Samuel L. Baker
QUEUING THEORY II 1 More complex queues: Multiple Server Single Stage Queue Queuing Theory II 2006 Samuel L. Baker Assignment 8 is on page 7. Assignment 8A is on page 10.  meaning that we have one line
More information1: B asic S imu lati on Modeling
Network Simulation Chapter 1: Basic Simulation Modeling Prof. Dr. Jürgen Jasperneite 1 Contents The Nature of Simulation Systems, Models and Simulation Discrete Event Simulation Simulation of a SingleServer
More informationPerformance Analysis of Sensor Networks by using finitesource
Performance Analysis of Sensor Networks by using finitesource queueing systems Târgu Mureş, 2015.09.02 1 2 3 4 In this paper we introduce a retrial queueing model to investigate the performance characteristics
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
More informationBig Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time
Big Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time Edward Bortnikov & Ronny Lempel Yahoo! Labs, Haifa Class Outline Linkbased page importance measures Why
More informationIntroduction to process scheduling. Process scheduling and schedulers Process scheduling criteria Process scheduling algorithms
Lecture Overview Introduction to process scheduling Process scheduling and schedulers Process scheduling criteria Process scheduling algorithms Firstcome, firstserve Shortestjobfirst Priority Roundrobin
More informationCS 4410 Operating Systems. CPU Scheduling. Summer 2011 Cornell University
CS 4410 Operating Systems CPU Scheduling Summer 2011 Cornell University Today How does CPU manage the execution of simultaneously ready processes? Example Multitasking  Scheduling Scheduling Metrics Scheduling
More informationIntroduction to Stationary Distributions
13 Introduction to Stationary Distributions We first briefly review the classification of states in a Markov chain with a quick example and then begin the discussion of the important notion of stationary
More information6.6 Scheduling and Policing Mechanisms
02068 C06 pp4 6/14/02 3:11 PM Page 572 572 CHAPTER 6 Multimedia Networking 6.6 Scheduling and Policing Mechanisms In the previous section, we identified the important underlying principles in providing
More informationOptimal Dynamic Resource Allocation in MultiClass Queueing Networks
Imperial College London Department of Computing Optimal Dynamic Resource Allocation in MultiClass Queueing Networks MEng Individual Project Report Diagoras Nicolaides Supervisor: Dr William Knottenbelt
More informationPERFORMANCE ANALYSIS OF SPEEDEDUP HIGHSPEED PACKET SWITCHES
PERFORMANCE ANALYSIS OF SPEEDEDUP HIGHSPEED PACKET SWITCHES Aniruddha S. Diwan, Roch A. Guérin, and Kumar N. Sivarajan Indian Institute of Science University of Pennsylvania Electrical Comm. Engg. Dept.
More informationCPU Scheduling. Prof. Sirer (dr. Willem de Bruijn) CS 4410 Cornell University
CPU Scheduling Prof. Sirer (dr. Willem de Bruijn) CS 4410 Cornell University Problem You are the cook at the state st. diner customers continually enter and place their orders Dishes take varying amounts
More informationPerformance Modeling and Analysis of a Database Server with WriteHeavy Workload
Performance Modeling and Analysis of a Database Server with WriteHeavy Workload Manfred Dellkrantz, Maria Kihl 2, and Anders Robertsson Department of Automatic Control, Lund University 2 Department of
More informationPerformance Workload Design
Performance Workload Design The goal of this paper is to show the basic principles involved in designing a workload for performance and scalability testing. We will understand how to achieve these principles
More informationSufficient Conditions for Monotone Value Functions in Multidimensional Markov Decision Processes: The Multiproduct Batch Dispatch Problem
Sufficient Conditions for Monotone Value Functions in Multidimensional Markov Decision Processes: The Multiproduct Batch Dispatch Problem Katerina Papadaki Warren B. Powell October 7, 2005 Abstract Structural
More informationLECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process
LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The
More informationQuantitative Analysis of Cloudbased Streaming Services
of Cloudbased Streaming Services Fang Yu 1, YatWah Wan 2 and RuaHuan Tsaih 1 1. Department of Management Information Systems National Chengchi University, Taipei, Taiwan 2. Graduate Institute of Logistics
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationProcess simulation. Enn Õunapuu enn.ounapuu@ttu.ee
Process simulation Enn Õunapuu enn.ounapuu@ttu.ee Content Problem How? Example Simulation Definition Modeling and simulation functionality allows for preexecution whatif modeling and simulation. Postexecution
More informationQueueing Networks with Blocking  An Introduction 
Queueing Networks with Blocking  An Introduction  Jonatha ANSELMI anselmi@elet.polimi.it 5 maggio 006 Outline Blocking Blocking Mechanisms (BAS, BBS, RS) Approximate Analysis  MSS Basic Notation We
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationPerformance Analysis, Autumn 2010
Performance Analysis, Autumn 2010 Bengt Jonsson November 16, 2010 Kendall Notation Queueing process described by A/B/X /Y /Z, where Example A is the arrival distribution B is the service pattern X the
More information