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

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 LIFO-push-out 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 LIFO-push-out 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 non-preemptive 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 teen-seconds 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 state-reduction 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 Bell-Northern 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 steady-state probabilities for infinite-state 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