Analysis of customers impatience in queues with server vacations

Transcription

1 Queueing Syst 26 52: DOI 7/s x Analysis of customers impatience in queues with server vacations Eitan Altman Uri Yechiali Received: 29 September 24 / Revised: November 25 C Science Business Media, LLC 26 Abstract Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival In this paper we consider systems with servers vacations where customers impatience is due to an absentee of servers upon arrival Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature We present a comprehensive analysis of the single-server, M/M/ and M/G/ queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline Keywords Queueing Single and multiple vacations Impatience Abandonment M/M/ M/G/ M/M/c Introduction Customers impatience has been dealt with in the queueing literature mainly in the context of customers abandoning the queue due to either a long wait already experienced, or a long This work was supported by the Euro-Ngi network of excellence E Altman INRIA, 24 route des Lucioles, BP 93, 692 Sophia Antipolis, France altman@sophiainriafr U Yechiali Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel uriy@posttauacil wait anticipated upon arrival Many authors treated the impatience phenomenon under various assumptions We mention a few of the works Palm s pioneering work [953, 957 seems to be the first to analyze queueing systems with impatient customers by considering the unlimited buffer M/M/c queue and assuming that each individual customer stays in the queue as long as his waiting time does not exceed an exponentially distributed impatience time Daley 965 studied the GI/G/ queue in which customers entering the system, if they are obliged to wait too long, may leave the system before starting or completing their service He derived an integral equation for the limiting waiting-time distribution function and investigated its solution for the cases of deterministic and of distributed impatience Takacs 974 further studied the M/G/ queue in which customers have a fixed threshold on their sojourn time, and derived the limiting distributions of the actual and of the virtual waiting times Baccelli et al 984 considered the GI/G/ queue where each aware customer, upon arrival, leaves immediately if he knows that his total waiting time is beyond his impatience threshold They devoted the analysis to characterization of waiting times in the system Boxma and de Waal 994 studied the M/M/c queue with generally distributed impatience times, developed several approximations for the abandonment probability and tested them via simulation Altman and Borovkov 997 considered impatience of customers in a retrial queue, in which a customer leaves the system if its commulative sojourn time exceeds some random threshold; impatience is shown to have an important impact on the system stability Van Houdt et al 23 presented an algorithmic procedure to calculate the delay distribution of impatient customers in a discrete time D MAP/PH/ queue, where the service time distribution of a customer depends on his waiting time They consider deterministic impatience time in the waiting room and in the system

2 262 Queueing Syst 26 52: The abandonment phenomenon and its importance with respect to stability of so called call centers has been studied extensively recently by various authors We direct the reader to the survey paper by Gans et al 23 and the many references there For the relevance of impatience in telecommunication systems, see eg Bonald and Roberts 2 However, there are situations where customers impatience is due to an absentee of servers upon arrival This situation is encountered, in particular, when observing human behaviour in service systems: if an arriving customer sees no server present in the system, he/she may abandon the queue if no server shows up within some time As a consequence, we analyze in this work queueing systems with servers vacations, where each arriving customer who finds no servers on duty, activates an independent random impatience timer If a server does not show up by the time the timer expires, the customer abandons the queue This timer procedure is different from the server s timer procedure studied by Boxma et al 22 and by Yechiali 24, in which the server, upon returning from a vacation to an empty system, activates a timer before taking another vacation We study both the multiple-vacation and the single-vacation models of the single-server M/M/ and M/G/ queues, as well as of the multi-server M/M/c systems The analyses of the M/M/ queues, as well as of the M/M/c systems, require the solution of a differential equation for the partial generating functions This is not a common occurrence when employing the partial generating functions method The analyses of the M/G/ queues are achieved by an interesting use of a related M/G/ model The M/M/c queues further require the calculation of certain probabilities, which are derived by finding the roots of a 2c-degree polynomial being the determinant of a certain matrix whose entries are functions of the system s parameters The paper consists of the following models: In Section 2 we consider the M/M/ queue with multiple server s exponentially distributed vacations and with exponentially distributed impatience times After deriving the balance equations we obtain and solve a differential equation for G z, the conditional generating function of the queue size when the server is on vacation This enables us to calculate the fractions of time the server is vacationing or busy In addition, we calculate P, the fraction of time the server is on vacation and the system is empty Section 3 deals with the multiple-vacation M/G/ queue where the vacation times, the service times and the impatience times are generally distributed We derive the Laplace-Stieltjes transform LST and the mean of the vacation period; the probability generating function PGF and mean of the number of customers at the start of a busy period; the LST and the mean duration of a busy period, and calculate P Furthermore, we derive the PGF of the number of customers at a service completion instant, and present a decomposition result Also, we have calculated the mean number of customers in the system at an arbitrary moment Finally, we derive a closed-form expression for Pserved, the fraction of customers served without abandoning the system Section 4 treats the multiple-vacation M/M/c queue with exponentially distributed vacation and impatience times Again, the balance equations lead to a differential equation for the PGF G z Interestingly enough, its solution is similar to the solution of the M/M/ case In order to obtain a complete solution for the unknown probabilities of the system state, the roots of a polynomial, obtained via the calculation of a square matrix, are determined Section 5, 6 and 7 are the counterparts of Section 2, 3 and 4, respectively, where the scenario is that of the server following the single-vacation service procedure 2 Multiple vacations: M/M/ queue with exponentially distributed vacation and impatience times 2 The model The underlying process is a M/M/ queue with multiple server vacations Levy and Yechiali, 975 The Poisson arrival rate is, service times B are exponentially distributed with parameter μ, and each vacation duration U is exponentially distributed with parameter Customers are impatient That is, whenever a customer arrives to the system and realizes that the server is on vacation he activates an impatience timer T, exponentially distributed with parameter, which is independent of the queue size at that moment If the server returns from his vacation before the time T expires and starts rendering service, the customer stays in the system until his service is completed However, if T expires while the server is still on vacation, our customer abandons the queue, never to return 22 Balance equations Let L denote the total number of customers in the system, and let J denote the number of working servers J = implies that the server is on vacation, while J = denotes that the server is active Then, the pair J, L defines a continuous-time Markov process with transition-rate diagram as depicted in Figure Let P jn = P{J = j, L = n} j =, ; n =,, 2, denote the steady state systemstate probabilities Then, the set of balance equations is given as follows: j = n = P = P μp n n P n = P,n n P,n, 2

3 Queueing Syst 26 52: Fig Transition-rate diagram j = n = μp = μp 2 P n 2 μp n = P,n μp,n P,n Define the Probability Generating Functions PGFs G z = P n z n, G z = P n z n n= 22 Then, by multiplying each equation for n in 22 by z n, summing over n and rearrange terms, we get G z[z μ z = zg z μp P z In a similar manner we obtain from 2 zg z = [ z G z P μp where G z = d dz G z We derive the solution of the differential Eq 24 in the following section 23 Solution of the differential equation Set A = P μp 25 Then, for z, G z [ z G z = A z Multiplying both sides of 26 by e z z d [ e z z / G z = A dz Integrating from to z we have e we get z z 26 e z z / G z G = A z s e s ds 27 s= Thus, G z = G e z z A e z z Then, z s= s e s ds 28 [ G = e G A [ lim z / z Since G =, we must have that G = A Define s= s= n= P def n s e s ds = P > and lim z / = z s e s ds 29 Z, := e Ɣ, Ɣ, where Ɣz is the Ɣ-function that has the representation Ɣz := t= exp tt z dt, and Ɣa, z := t=z exp t t a dt Further define K := s= s e s ds Some computations give K = Z, 2 We then write, using 25, G = P = P μp K = K μ K P 2 It is easy to check that K > Indeed, K = s/ e s ds < s/ ds = Now, substituting in 28 the value of A from equation 29 we

4 264 Queueing Syst 26 52: obtain G z = G e z [ z s / e s ds s, e s ds z / 22 Using L Hopital rule, the probability that the server is on vacation, P = n= P n, is derived: G = P = G s e s ds = K G = K P 23 Clearly, the probability that the server is working is P = P n = P By substituting G from 2 we get the relation G = P = Clearly, the above expression, A K K = A = P μp P = P μp 24 can be obtained by considering a horizontal cut between the two levels of the transition rate diagram Figure Equation 22 expresses G z in terms of G = P, the proportion of time the server is on vacation and there are no customers in the system Also, G z is a function of G z and P Thus, once P is calculated, G z and G z are completely determined, P is obtained from 2, and P is given by 23 Finally, the proportion of customers abandoning the system is PT < UP = P Nevertheless, the necessary and sufficient condition for stability is <μ, for otherwise the busy period has an infinite expectation We derive the value of P in the next section 24 Derivation of P, P, P, E[L and of E[L From 23, G z = [ G z μp P z z μ z Applying L Hopital rule, we get G = [ G μp P G μ where G E[L = np n, Since G j = P j j =, 2, by applying equation 24 to the numerator of G above, we have E[L = μ P 25 On the other hand, from equation 24, E[L = lim G z = G G z = P E[L, implying that E[L = P 26 Equating the two expressions 25 and 26 for E[L, and using = P P,weget P = μ μ, P μ = μ μ, 27 implying that E[L = μ μ μ Now, using equation 23, we finally obtain P = K P = K μ μ μ, 28 where K is given by equation 2 P [P is a decreasing [increasing convex [concave function of, having its limits at [at /μ when, μ and at [at, respectively when E[L behaves similar to P Indeed, E[L when, and μ E[L when that is, every arrival who finds the server on vacation leaves immediately As for P, its behavior as a function of is given in Figure 2 for parameter values =, μ = 2, and = In general, it is an increasing concave function of with asymptote at since, as, the probability that at the end of a vacation the system is non empty, converges to zero When then P approaches 25 see also section 3 Derivation of E[L From 23 we have G z = G zz Az z μ z,

5 Queueing Syst 26 52: P The second term above follows since a new arrival does not change the sojourn time of a customer present in the system, while the third term takes into account the probability n/ n that, when there is an abandonment among n waiting customers, our customer will not be the one to leave Thus, n E[S n = n n n μ n E[S,n 22 Fig xi P as a function of By using L Hopital rule and P = A, we derive E[L = lim z G z = E[L L 2 E[L 2μ 2, where E[L L is obtained by differentiating twice G z atz = We also have E[S = μ E[S implying that E[S = μ, 25 Sojourn times Let S be the total sojourn time of a customer in the system, measured from the moment of arrival until departure, either after completion of service or as a result of abandonment By Little s law, E[S = E[L = E[L E[L 29 However, a more important measure of performance is S served, defined as the total sojourn time of a customer who completes his service Let S jn denote the conditional sojourn time of a customer who do not abandon the system, given that the state upon his arrival is j, n Clearly, E[S n = n, n =, 2, 3, μ When J =, for n, E[S n = n n E[S n n n E[S n n n n n n E[S,n Iterating 22 we obtain, for n, E[S n = [ n k n k= k n j j=k j n n n μ Finally, using the expression for E[S n, we write k μ 22 E[S served = P n E[S n P n E[S n 222 That is, E[S served = E[L P μ n= P n E[S n 223 n= One might be interested in the conditional expectation of W, n, the total sojourn time of a customer in the system, regardless if served or not, given that upon arrival he observes the state, n Then, for n, E[W, n = n n E[S n

6 266 Queueing Syst 26 52: n E[W, n n n n n E[W, n, n and, for n =, E[W, = μ E[W, After calculations, the above yields E[W, n = n k= n! n k k! μ μ k n j=k j 3 Multiple vacations: M/G/ queue with generally distributed vacation and impatience times In this section we consider the case of generally distributed service times, ie, the underlying process is the M/G/ queue with multiple server vacations cf Levy and Yechiali 975 The arrival process is Poisson with rate Service times are iid random variables, all copy of B, having first moment E[B, second moment E[B 2 and Laplace Stieltjes transform LST B s = E[e sb At the end of a busy period the server takes a vacation U, having finite moments E[U and E[U 2, and LST U s If the system is empty at the end of a vacation, the server takes another vacation If there are n customers at the end of a vacation, the server starts immediately a busy period When the server is on vacation and is not available for service, arriving customers are impatient An arrival who finds that the server is away on vacation, activates an impatience timer, T If the server does not return by the time T, the customer abandons the system Each customer activates its own timer and the T i s are iid random variables, independent of the number of customers waiting Let the starting time of a vacation be t = Then, a key observation is that, within U, the evolution of the system is the same as that of a M/G/ queue with service times all distributed as T For time t U, it is well known Takacs, 962 that the number of customers in the system has a Poisson distribution with parameter t = t [ PT ydy, t U 3 We will use this observation extensively in the sequel 3 Duration of a vacation period, τ Consider the time t = when the server first leaves for a vacation of duration U If at time t = U the queue is empty, the server takes another vacation U 2, and so on This sequence of events terminates at the first time when the server returns and finds a non-empty system We call this entire length of time, τ, a Vacation Period Using the M/G/ analogy, the probability of an empty system at time U is e U Thus, τ = k U i U k with probability i= e U k k i= e U i Therefore, the LST, τs, of the Vacation Period is given by τ s = E e s k U i i= e su k e k i= = k= U i e U k [ E e su U k [ E e su E [ e su U k= = U s E [ e su U E [ e su U 32 It follows that the mean length of a Vacation Period is E[U E[τ = E [ e U Number of customers at a start of a busy period A busy period starts with Nτ customers We now derive the Probability Generating Function PGF of Nτ It should be pointed out that Nτ isnot distributed as a Poisson variable with parameter This follows since the last vacation U in τ in which there is at least one arrival is not a regular one Indeed, U s NU = E [ e su NU = E [ e su I {NU } E[PNU = U s E [ e su U E [ e U

7 Queueing Syst 26 52: This results in We write E [U NU { NU if NU Nτ = N τ if NU = = E[U E [ Ue U E [ e U 34 is given by Ɣ s = E[e sɣ = E[e s Nτ θ i i= = [θ s Nτ = G Nτ θ s Using 35 we get Ɣ s = E[e θ s U E[e U 38 E[e U where N τ and τ are iid replicas of Nτ and τ, respectively Then, the PGF of Nτisgivenby G Nτ z = E[z Nτ = E { E[z NU NU PNU } E { E[z Nτ NU = PNU = } [ = E z n U Un e n! Thus, E[z Nτ E[e U G Nτ z = E[e z U E[e U E[e U n! E[e U U n z n = 35 E[e U In particular, with ρ = E[B, E[Ɣ = E[NτE[θ = E[ U E[e U E[B ρ 39 Now, the proportion of time the server is busy, P busy,is give by P busy = E[Ɣ E[Ɣ E[τ = E[ UE[B E[ UE[B ρe[u 3 Indeed, for the M/M/ case, P busy = P as given by 27 Furthermore, P busy <ρ To see this, it is enough to show that E[ U <E[U In fact, [ U [ U E [ PT y dy E dy = E[U 3 It readily follows that PNτ = n = and E[Nτ = n! E[e U U n E[e U n =, 2, 3, 36 E[ U E[e U 33 The busy period and P busy 37 Let Ɣ denote the duration of a busy period A busy period starts with Nτ customers, and hence is equal to the sum of Nτ iid regular M/G/ periods θ, θ 2,,θ Nτ, all distributed like θ, where θ s = B [s θ s Kleinrock, 975, p 22 Thus, the LST of Ɣ 34 Calculation of P We now calculate P, defined in section Let D denote the sum of time intervals, within τ, where the system is empty That is, D = τ I {Nt = }dt 32 Due to the regenerative property of the system we can write D = U I {Nt = }dt D I {NU = } where D has the same distribution of D Since E[I {Nt = } = e t we have E[D = E[ U e t dt E[e U 33

8 268 Queueing Syst 26 52: Now, since P is the fraction of time in which both the system is empty and the server is on vacation, we have P = E[D E[Ɣ E[τ 34 Using 33, 3 and 33 we finally get where the last equality comes from equation 2 Thus, P = ρ K ρ ρ = K μ μ, which is the expression for P given by eq 29 E[ U P = e t dt E[ UE[θ E[U E[ U = ρ e t dt E[ UE[B ρe[u Number of customers at a service completion instant Let X n denote the number of customers present after the completion of the n-th service We have For the case where the impatience variable T is exponentially distributed with parameter, P = [ U ρe e e t dt ρ [ U ρe[u 36 Note that when we get: P = ρ E[U U If U is exponentially distributed with parameter then this simplifies to P = ρ E[U = ρ In particular, if we substitute the parameters values used in Figure 2 we obtain 25, which indeed coincides with what we see in the figure 34 Exponentially distributed vacation and impatience times Supposing that U Exp, equation 36 yields ρ u= e u u t= e e t dt du P = [ ρ ρ By changing order of integration and applying change of variable: s = e t in the numerator above, we get X n = d { Xn AB if X n Nτ AB if X n = 37 where At is the number of Poisson arrivals in, t The symbol = d means equal in distribution Therefore, in steady state, the PGF of X = lim X n is given by n ˆXz = E[z X = E[z X X > z E[z AB P where P = X = Thus, E[z Nτ z E[z AB P = z B [ z [ ˆXz P E[z Nτ P ˆXz = P E[z Nτ z B [ z B [ z 38 where E[z Nτ = G Nτ z is given by 35 To calculate P we substitute z = in 38 and apply L Hopital rule to get P = PX = = ρ E[Nτ = ρ E[e U E[ U 39 where ρ = E[B and E[Nτ is given by 37 Note that P P since P is the fraction of time the system is empty, while P is the relative frequency of occurrences, among service completion instants, when the system becomes empty Finally, s= e s s / ds s = s= s e s ds = K, ˆXz = ρ G Nτz B [ z E[Nτz B [ z E[e zu B [ z = ρ E[ Uz B [ z 32

9 Queueing Syst 26 52: When T =, the process transforms into the classical M/G/ queue with multiple server vacations Then, U = U and G Nτ z /E[Nτ = U [ z / E[U Equation 32 then leads to the known expression see Levy and Yechiali 975 and Boxma, et al 22 U ˆXz [ z = ρ EUz B [ z B [ z Mean number of customers at an arbitrary moment We now calculate E[L, the mean number of customer in the system at an arbitrary moment We write E[L = E[L Vacation Period P busy E[L BusyP busy 324 Consider a vacation period τ Let Nt be the number of customers in the system at time t [,τ Let = τ Ntdt Then, 36 A decomposition representation Equation 32 can be written in a decomposition form, namely, ˆXz = ˆL M/G/ z G Nτ z z E[Nτ 322 where ˆL M/G/ z is the PGF of the system s state occupancy at an arbitrary moment in the corresponding regular M/G/ queue, given by ˆL M/G/ z = ρ z B [ z z B [ z That is, X is the sum of two independent random variables, L M/G/ and Y, where the PGF of Y is given by Ŷ z = G Nτz z E[Nτ Note that, because of customer abandonments due to impatience, one cannot use PASTA and Burke s theorem to deduce that the distribution of X, the number of customers at service completions, is the same as the distribution of L, the number of customers at an arbitrary instant Now, from 322 and using 35, it follows that E[X = E[L M/G/ E[Y = [ 2 E[B 2 2 ρ ρ E[ U2 2E[ U 323 = U Ntdt I {NU = } where has the same distribution as, and Nt has a Poisson distribution with parameter t Taking expectations we get [ U E Ntdt E[ = E [ e U Then, using 33, [ U E[L Vacatoin Period = E[ E E[τ = Ntdt E[U Now, E [ U Ntdt = E [ t, implying that E[L Vacation Period = E [ U tdt E[U 325 In particular, when U Exp and T Exp, then E[L Vacation Period = Now consider a busy period which starts with Nτ customers Then, E[L Busy [ Nτ = E Nτ {E[L M/G/ Busy Nτ n} = E[NτE[L M/G/ Busy 2 E Nτ[NτNτ 326 When T, E[ U2 is the residual life time of U 2E[ U = E[U 2 2E[U = E[R U, where R U Clearly, E[L M/G/ Busy = E[L M/G/ /ρ Collecting terms, we obtain

10 27 Queueing Syst 26 52: [ U ρ E[L = E tdt ρe[u E[ UE[B {E[NτE[L M/G/ /ρ 2 } E[NτNτ E[ UE[B ρe[u E[ UE[B 327 where E[Nτ is given by 37, E[L M/G/ is given by the first term in the right hand side of 323 and E[NτNτ = E[ U2 E [ e U, 328 which is derived by differentiating 35 twice at z = In particular, when T Exp, and U Exp, then [ i E[ U 2 = 2 2 2, 2 ii E[e U = K, iii E [ U = tdt = [ E[U Substituting the above into 327, together with E[ U = / which we shall establish in 33, yields an explicit solution for E[L when T and U are distributed exponentially Further note that it can readily be shown that in the exponential case, and with E[B = /μ, the first term in 327 coincides with E[L given in Section Proportion of customers served An important performance measure is the proportion of customers served, denoted by Pserved We can write, Pserved = Expected number of customers served during a cycle Expected number of arrivals during a cycle 329 Using 3 and then 33 and 37, we get Pserved = E[Ɣ/E[B [E[Ɣ E[τ = P busy ρ = E[U ρ ρ E[ U 33 Clearly, Pserved when ρ For exponentially distributed vacations and impatience times, where U Exp and T Exp, we have U U = e y dy = e U y= leading to E[ U = It follows that Pserved = u= e u e u du = 33 ρ ρ = ρ Multiple vacations: M/M/c queue with exponentially distributed vacation and impatience times 4 The model We consider a M/M/c type queue with c server and multiple vacations see Levy and Yechiali 976 and recent studies using matrix geometric methods by Zhang and Tian 23 Service time, B, of each individual customer is exponentially distributed with mean /μ The arrival process is Poisson with rate, and we assume that <cμ Servers never stay idle in the service station: when a server finishes a service and finds no waiting customers in the queue, he immediately leaves for a vacation The duration of a vacation by a server is exponentially distributed with mean / If there are j c customers attended by j servers and one of the c j vacationing servers returns from a vacation to find an empty queue, he immediately leaves for another vacation Customers are impatient: an arriving customer who finds that all servers are on vacation activates an independent timer, with exponentially distributed duration, having mean / If no server becomes available by the moment the timer expires, the customer abandons the queue and the system 42 Balance equations Let L denote the total number of customers in the system, and let J denote the number of working servers Then the pair J, L defines a continuous-time Markov process with stationary probabilities P jn = P{J = j, L = n} j c, n j The set of steady state balance equations for this

11 Queueing Syst 26 52: process is given as follows: n = P = μp P j = n n c P n 4 = P,n n P,n = [ c j z j jμz j P jj c j z j P j, j z j μp j, j z j = z j [ c j z jμ P jj c j P j, j j μzp j, j 46 n = j j c n > j n = c n = c k j = c jμp jj = jμp j, j c j P j,n j μp j, j 42 [ jμ c j P jn = P j,n jμp j,n c j P j,n cμp cc = cμp c,c P c,c cμp c,ck = P c,ck cμp c,ck Finally, from 43, for j = c, weget [z z cμ zg c z zg c z = z c [ cμp cc P c,c 47 Define, for every j, the marginal probability P j = n= j P jn = ProbJ = j Then, by substituting z = in each of the equations 45, 46 and 47, and repeated substitution, we get, for j c, c j [P j P jj = j μp j, j Solution of the differential equation for G z k =, 2, 3, P c,ck 43 Interesting enough, the differential equation 45 is similar to equation 24, where the only difference is that the term c replaces the term Therefore, the solution is given by see Generating functions For every j j =,,,c we define a partial Generating Function GF and its derivative: G j z = P jn z n G j z = d dz G jz 44 n= j By multiplying by z n each equation for j and n in 4, 42, and 43, and summing over n we obtain the following: From 4, for j =, we derive a differential equation for G z: zg z = [ z c G z c P μp 45 From 42, for j c, we obtain a set of linear equations in G j z: [z jμ z c j zg j z c j zg j z G z = P e z z c z c z s= A c e z s c e s ds 49 where, similarly to the single server case, A c = c P μp, and P = G = A c s c e s ds s= def = A c K c 4 K c can be computed using 2 with c replacing Thus, P = A c K c = c P μp K c, implying that A c = P /K c and c K c P = μk c P 4 In order to obtain G z completely we need to calculate P This is accomplished in section 47, together with the calculation of all P jj and P j

12 272 Queueing Syst 26 52: Evaluating the mean busy period duration, E[Ɣ c In the M/M/c with vacations model, when all servers are on vacation J =, the equivalent of a single vacation U of the M/G/ queue is the variable V, distributed exponentially with parameter c Thus, the duration of a Vacation Period is τ c = k k V i V k with probability i= e V k i= e V i Therefore, similarly to equation 32, the LST of τ c is given by Set a j z = z jμ z c j z j c and 46 b j z = z j [ c j z jμ P jj c j P j, j j μzp j, j j c b c z = z c [ cμp cc P c,c Then, equations 46 and 47 can be written as 47 τ c s = V s E[e sv V E[e sv V 42 a j zg j z c j zg j z = b j z j c 48 where V s = E[τ c = c Also, similarly to 33, c s E[V E[e V = [ c E[e V 43 The PGF, probability mass function and mean of Nτ c are given, respectively, by equations 35, 36 and 37, where V Expc replaces U The equivalent of D Section 34 is a random variable D c, with mean similarly to 33, E[D c = E [ V e t dt E[e V 44 The busy period, Ɣ c, is defined as the length of time from the moment, starting at τ c, when one of the c vacationing servers returns from a vacation to find Nτ c waiting customers, until the first moment thereafter in which there are no customers in the system anymore Thus, as in 34, P = E[D c E[Ɣ c E[τ c 45 Once P is calculated, E[Ɣ c is directly determined from Solution of the set of generating functions The set of equations 46 and 47 is a set of c linear equations with c variables G j z, j =,, 2,,c, where G z is given by 49 and the c unknowns are the PGFs G j z, j =, 2,,c where the terms b j z are functions of the probabilities P jj for j =, 2,,c Define a c c square matrix Qz as Qz a z c z a 2 z c 2 z a 3 z c 3 z a 4 z = 2 za c z z a c z Set the vector d T z = d z, b 2 z, b 3 z,,b c z, where d z = b z c zg z Also, set g T z = G z, G 2 z,,g c z Then, the set 48 can be written in a matrix form as Qzgz = dz 49 To obtain G j z we use Cramer s rule and write Qz G j z = Q j z, j c 42 where Q denotes the determinant of the matrix Q, and Q j z is a matrix obtained from Qz by replacing the jth column with the rhs vector of 49, dz It thus follows that the functions G j z are expressed in terms of the c probabilities P jj, j c Once P is calculated, P is extracted from 4, but we still have to find c additional

13 Queueing Syst 26 52: equations for the unknowns P jj The following theorem answers this requirement Theorem The 2c-degree polynomial Qz has exactly c distinct roots in the interval, Proof: Qz = c j= a jz For j = c, a c z = zz cμ, implying that the two roots of a c z are z c = and z c 2 = cμ/ > For j c, a j z = z 2 jμ c j z jμ, implying that a j = jμ <, a j = c j >, while a j = Thus, the two roots of a j z are < z j < and < z 2 j < Clearly, all z j roots for j c are distinct, which completes the proof Now, since G j z is positive for < z <, then Q j z vanishes whenever Qz does Thus, each equation Q j z j =, j =, 2,,c, yields an independent equation in the probabilities P jj, j c Once P and P are known, the values of the required P jj for 2 j c are directly obtained Given P jj for j c, the PGFs G j z are completely determined by equation Simultaneous calculation of P jj, P j and G j E[L j for j c The calculation of P for the multiple-vacations M/M/c queue is more elaborate than the calculation in the multiplevacation M/M/ case Indeed, P is derived simultaneously with all unknowns P jj and P j as follows Equation 4 gives a relation between P and P, namely, c K c P = μk c P Similarly to the derivation of equation 23, by using equations 49 and 4 we get P = c K c P 42 This implies, similarly to 24, that c P = c P μp = A c 422 Also, from 45, and similarly to 26, we obtain E[L = P c 423 where E[L j G j = np jn j =,, 2,,c 424 n= j From 48, after taking derivatives at z =, we have, for j c a j P j a j G j [c j P j G j = b j 425 where, from 46, a j = jμ c j j c, 426 and, from 47, for j =, 2,,c, b j = j [ c j jμp jj c j P j, j while, for j = c, j μp j, j c j Pjj j μp j, j 427 b c = c[ cμp cc P c,c 428 Thus, there are 3c unknowns involved: P jj, P j and E[L j = G j, for j =,, 2,,c Their values are obtained by solving a set of 3c linear equations as follows: Relations 4, 42 and 423 give 3 independent equations involving P, P, P and E[L The set 425 yields another c independent equations relating the unknowns P j for j c and E[L j for j c There are additional c independent equations resulting from the Theorem in section 46 The set 48 adds c equations relating to the P jj s and the P j s The last equation is c j= P j = All in all we have 3 c c c =3c 3 independent equations to solve for the 3c unknown variables 5 Single vacation: M/M/ queue with exponentially distributed vacation and impatience times 5 The model We consider now the M/M/ queue where the server takes only a single vacation see [Levy and Yechiali, 975 at the end of a busy period If the server returns from a vacation to an empty system he waits dormant to the first arrival thereafter, who opens a busy period Otherwise, if the queue is positive at the vacation s termination, the server starts a busy period with no delay Customers are, as before, impatient when they find, upon arrival, that the server is vacationing Each customer activates his independent, exponentially distributed impatience time, T

14 274 Queueing Syst 26 52: Balance equations and PGFs The system state is, as before, J, L Figure can represent the transition rate diagram of the single vacation case if we add the state, to it The balance equations are now n = P = P μp j = n n P n 5 = P,n n P,n n = P = P j = n μp n 52 = P,n μp,n P n Summing 52 over n we obtain, similarly to 24, that in this case P = μp Define the PGFs: G j z = n= P jnz n, j =, 2 Then, similarly to section 22, we obtain from 5 G z zg z = zg z G z μp and from 52 G z μg z P 53 = zg z μ z G z P z P G z 54 Equation 53 can be written as zg z = [ z G z μp 55 implying z = that, indeed, P = μp 56 The differential Eq 55 is very similar to equation 24 and therefore its solution is given by equation 22 with G = P = μp K see 2 Using the last relation, as well as P = P from 52, equation 54 is written as G z[z μ z = zg z μ z K z P 57 Also, from 57, when z = we get, as in 23, P = K P 58 Again, once P is calculated, both G z and G z are completely determined, as well as P, which is given by 23 We ll calculate P in the next section 53 Calculation of P From equation 57, G z = zg z μ z K z P z μ z Applying L Hopital rule, we get G = G G μ μ That is, K P 59 G = E[L = μ μ P P P K 5 On the other hand, from 55 E[L = lim G z = G G, z implying that E[L = P 5 Equating the two expressions 5 and 5 for E[L, using equation 58 and = P P, we obtain the needed expression for P by which P and E[L are obtained: [ P K μ K μ = μ 52 Note the similarities and differences between 52 and 28 Clearly, P single vacation < P multiple vacation 54 Sojourn times Let S and S jn be defined as is section 25 Then, E[S is given by expression 29, with E[L derived from 59 E[S n = n /μ, but this time for n =,, 2,rather than for n E[S n is given by 22 Finally, E[S served is given by 222 and 223, but with the first sum in 222 starting from n =

15 Queueing Syst 26 52: Single vacation: M/G/ queue with generally distributed vacation and impatience times The setting is as in Section 3, but under the single-vacation policy 6 The busy period A busy period starts either with NU = n customers with probability e U U n /n! or with a single customer with probability e U Thus, the LST of the busy period is [ Ɣ s = E θ s n e U U n /n! E [ θ se U = E [ e U e Uθ s θ se [ e U = E [ e U θ s θ se [ e U 6 From 6 it follows that E[Ɣ = E[θ E[ U E [ e U 62 where E[θ = E[B/ ρ 62 Calculation of P, P and P A cycle C consists of a busy period Ɣ, a single vacation U, and, with probability E [ e U, an exponential inter-arrival time with mean / Thus, E[C = E[Ɣ E[U E [ e U / 63 In the single vacation case the sum D of time intervals, within a vacation U, in which the system is empty is given by D = and U [ U E[D = E I {Nt = }dt 64 e t dt 65 The fraction of time in which there are no customers in the system and the server is on vacation is, using 65, 63 and 62, P = E[D E[C = ρ E [ U e t dt E[B E[ U E [ e U ρ E[U E [ e U/ 66 while the fraction of time in which the server is ready to serve but there are no customers present in the system is P = E [ e U/ E[C Finally, P = E[Ɣ E[C 67 Remark Exponentially distributed service, vacation and impatience times In the case where B Expμ, U Exp and T Exp, similarly to the derivation in Section 34, [ U E e t dt = K t=, E [ e U = K Using 33, it follows that eq66 reduces to Number of customers at a service completion instant Define X n and P as in section 35 Then, for a given U, 68 X n = d X n AB if X n NU NU AB if X n =, NU AB NU= if X n =, NU = Thus, in steady state, the PGF of X is given by ˆXz = P E[z X X > z B [ z P E[E[z NU NU > z B [ z PNU > e U B [ z = ˆXz P z B [ z P E{[ ˆNU e U z B [ z e U B [ z} 69

16 276 Queueing Syst 26 52: That is, U ˆXz [ z ze[e U = P z B [ z B [ z 6 P is given by ρ P = E[U E [ e Finally, U 6 U ˆXz [ z ze [ e U = ρ [ E[U E e U z B [ z B [ z Proportion of customers served The proportion of customers served without abandoning the system is defined as in equation 329 Then 7 Single vacation: M/M/c queue with exponentially distributed vacation and impatience times 7 The Model In this section we consider an M/M/c-type queue, similar to the one described in section 4, but with each server taking only a single vacation at a time 72 Balance equations As in section 42, the pair J, L defines a continuoustime Markov process where, now, for every j, the state space consists of all n for every j, rather than only of n j Specifically, let P jn = P{J = j, L = n} j c; n Then, the set of balance equations is given as follows: n = c P = μp P j = n n c P n = P,n 7 n P,n Pserved = E[Ɣ/E[B E[C Using 62 and 63 we have Pserved = = P busy ρ E[ U E [ e U [ ρ ρ [ ρ E[ U E e U E[U E [ e U E[ U E [ e U = ρ E[ U E [ e U ρe[u 63 Again, when ρ, then Pserved For U Exp and T Exp, we have from 33 that E[ U =, and from 68 that E [ e U = K Thus, Pserved = K ρ K ρ Comparing 64 with 332, it follows that 64 Pserved single vacation > Pserved multiple vacations j c n = n = j [ c j P j = c j P j, μp j, n j [ nμ c j P jn n > j = P j,n c j P j,n n μp j,n [ jμ c j P jn = P j,n c j P j,n jμp j,n j μp j,n [ jμ c j P jn = P j,n c j P j,n jμp j,n 72

17 Queueing Syst 26 52: j = c n = n c nμp cn = P c,n P c,n n c 73 Generating functions P c = P c, cμp cn 73 = P c,n P c,n cμp c,n Similarly to section 43 multiple vacation case we define, for every j c, the partial Generating Function G j z and the marginal probability P j = G j, where G j z = P jn z n 74 n= By multiplying by z n and summing over n we obtain: From 7, for j = : zg z = [ z c G z μp 75 From 72, for j c [z jμ z c j zg j z c j zg j z = jμ z μz j P jn z n c j zp j, n= j j np jn z n μ np j,n z n n= From 73, for j = c 76 [z cμ zg c z zg c z = c c μz np cn z n cμ z P cn z n cμp cc z c 77 n= The set 76 and 77 can be written in a matrix form, similarly to 49, as Qzgz = dz where Qzis exactly the same matrix defined in section 46 with the same a j z for j c, and d T = d z, b 2 z, b 3 z,, b c z, where, as before, d z = b z c zg z, but, b j z = jμ z μz j P jn z n c j zp j, n= j j np jn z n μ np j,n z n n= j c and c c b c z = μz np cn z n cμ z P cn z n n= 78 cμp cc z c Solution of the differential equation and calculation of the unknown probabilities The differential equation 75 is similar to the differential equation 45, implying that the former s solution is given by equation 49 with the only modification that the term μp replaces the term A c = c P μp That is, G z = e z z c P μp z s c e s ds 7 s= Furthermore, similarly to 4 P = μp s c e s ds = μp K c 7 From equations 7 and 7, similarly to the derivation of 23 and 42, P = c K c P = μ c P 72 which can also be obtained directly from 75 by setting z = By setting z = in equations 73 and 77 we get, respectively, for j c c j P j c j P j, = c j P j, μ j np jn j μ np j,n 73

18 278 Queueing Syst 26 52: and for j = c c P c P c, = μ np cn cμp cc 74 Equations 72, 73 and 74 comprise a set of c equations involving the c marginal probabilities P j j c and the boundary probabilities {P jn } for j c, n j, excluding P From 75, letting z and using L Hopital s rule, we get E[L = G c G = P c 75 Since equation 48 holds here too although with different b j z, see 78 and 79, equations 425 and 426 hold here also but, instead of 427 and 428 we have, respectively, b j and = jμ j μ n= P jn c j P j, j np jn μ j j n 2 P jn μ n 2 P j,n 76 c b c = μ np cn μ c c n 2 P cn cμ P cn 77 n= The unknown probabilities are {P jn } for j c, n c; {P j } for j c; and E[L j for j c Altogether there are c c /2 2c = c 2 7c 6/2 unknowns They are solved by the combination of equations 7, 72, 75; 425 with the updated b j ; c equations are derived from the roots of Qz; equations 73 and 74; c c/2 balance equations for the boundary probabilities {P jn }, where j c, n j ; and the total probability equation c j= P j = That is, there are c c c c c/2 = 3c 3 c c/2 = c 2 7c 6/2 equations Compared with the M/M/c multiple vacation case, the solution for the single vacation model requires additional c c/2 equations for the unknown probabilities {P jn } where j c, n j 8 Conclusion We have introduced and analyzed in this paper a new type of impatience behavior in which customers become impatient and may leave the system when the server goes on vacation This is in contrast with previously studied impatience behavior which did not consider server vacations and where customers may become impatient when the number of customers or the amount of workload queued in front of them is large We analyzed both the single and the multiple vacations cases We studied both Markovian models the M/M/ and the M/M/c queues with exponentially distributed vacation and impatience times as well as the M/G/ case with generally distributed impatience and vacation times For the M/M/ case, we derived explicit expressions for the PGF of the number of customers conditioned on the server state in the system For the M/G/ model, we obtained the PGF of the number of customers at various embedded instants end of service, start of a busy period; we calculated the mean number of customers in the system at an arbitrary moment, and we derived other performance measures, including the proportion of customers being served For the M/M/c queue we derived explicit expressions for the PGF of the number of customers conditioned on the server state in terms of several constants, which are derived by finding the roots of a 2c-degree polynomial being the determinant of a certain matrix whose entries are functions of the system s parameters In many queueing problems, working with the PGF of number of customers in the system or with the LST of the workload allows one to transform difference equations resp differential equations that represent the balance equations for the steady state probabilities, into algebraic equations Interestingly, in the Markovian models we introduced in this paper, the difference equations describing the balance equations does not transform into algebraic equations for the PGF: the PGF is characterized by a solution of a differential equation which we solved explicitly for each model References Altman, E and Borovkov, A A, On the stability of retrial queues, Queueing Systems, Baccelli, F, Boyer, P and Hebuterne, G, Single-Server Queues with Impatient Customers, Advances in Applied Probability, Bonald, T and Roberts, J, Performance modeling of elastic traffic in overload, ACM Sigmetrics, Cambridge, MA, USA, Boxma, OJ and de Waal, PR Multiserver Queues with Impatient Customers, ITC, Boxma, OJ, Schlegel, S and Yechiali, U, A Note on the M/G/ Queue with a Waiting Server, Timer and Vacations, American Mathematical Society Translations, Series 2, Daley, DJ, General Customer Impatience in the Queue GI/G/, J Applied Probability, Gans, N, Koole, G and Mandelbaum A, Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, Kleinrock, L 975, Queueing Systems Volume I: Theory, J Wiley & Sons, New York

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