A QUEUE-LENGTH CUTOFF MODEL FOR A PREEMPTIVE TWO-PRIORITY M/M/1 SYSTEM

Transcription

1 A QUEUE-LENGTH CUTOFF MODEL FOR A PREEMPTIVE TWO-PRIORITY M/M/1 SYSTEM QIANG GONG AND RAJAN BATTA Abstract We consider a two-priority preemptive single-server queueing model Each customer is classified into either a high priority class or a low priority class The arrivals of the two priority classes follow independent Poisson processes and service time is assumed to be exponentially distributed A queue-length-cutoff method is considered Under this discipline the server responds only to high priority customers until the queue length of the other class exceeds a threshold L After that the server switches to handle only the low priority queue Steady-state balance equations are established for this system Then we introduce two-dimensional generating functions to obtain the average number of customers for each priority class We then focus on the preemptive resume case while allowing for weights associated with both priority class queues We develop methodologies to obtain the optimal cutoffs for the situation when the weights of both queues are constant (ie not a function of queue length) and the situation when the weights change linearly with the queue lengths It is important to point out that our method does not lead to a closed-form exact solution but rather a numerical approximation from which cutoff policies are analyzed Key words priority queue queue-length cutoff generating function 1 Introduction and Literature Review Our research is primarily motivated by a disaster-relief project which deals with how to rescue casualties after a disaster occurs We consider a dynamic disaster environment (eg earthquake) in which thousands of casualties need to be treated The casualties in such a disaster setting are usually placed into four levels (see the description of HAZUS a GIS-enabled software used by the Federal Emergency Management Agency FEMA for the purpose of earthquake loss estimation in the paper by Al-Momani and Harrald [1]): 1 Severity level 1 - injuries will require medical attention but hospitalization is not needed 2 Severity level 2 - injuries will require hospitalization but are not considered life threatening 3 Severity level 3 - injuries will require hospitalization and can become life threatening if not promptly treated 4 Severity level 4 - victims are killed by the earthquake In an earthquake disaster relief setting (eg the one that occurred in Northridge CA in 1994) severity level 1 and 4 calls are initially not responded to Thus the system operates as a two-priority queue with severity level 3 being priority 1 and severity level 2 being priority 2 Since injuries can rapidly deteriorate when unattended it is possible that severity level 2 injuries that are left unattended for a long period of time can become even more critical than a typical severity level 3 injury Thus operating in a strict priority queue model in a heavy-traffic situation would be detrimental This provides the motivation to study a two-priority queueing system with a queue-length cutoff This cutoff model is being implemented in the software for disaster-relief being developed at the Center for Multisource Information Fusion at the University at Buffalo (SUNY) Details of its effectiveness via case studies developed for an earthquake scenario in Northridge CA will be presented in a later paper Enterprise Optimization United Airlines 1200 East Algonquin Elk Grove Township IL Bell Hall Department of Industrial and Systems Engineering and Center for Multisource Information Fusion University at Buffalo (SUNY) Buffalo NY Author for correspondence (batta@engbuffaloedu) 1

2 2 There are other applications of this queue-length cutoff model For example telecommunication in ATM (asynchronous transfer mode) networks also has this flavor Voice data must flow through the network without noticeable distortion or delay Losing a chunk of voice data isn t a problem but delay or receiving it out of order is So voice is delay sensitive loss insensitive On the contrary computer data are delay insensitive loss sensitive since individual chunks are not of much use until they are all received but in many cases data delay in transmission is often acceptable Based on the characteristics of both types of data voice is classified as high priority class while computer data as low priority class Again computer data cannot be indefinitely delayed so it makes sense to have a queue-length cutoff model in such a situation Previous research in the area of priority queueing models may be categorized as either server cutoff or queue-length cutoff Figures 11 and 12 illustrate two straightforward examples for both types of models respectively High priority queue Server 1 Server 2 N high Low priority queue Server 3 Server 4 N low Fig 11 Server cutoff model High priority queue Server 1 Server 2 L Low priority queue Server 3 Server 4 Fig 12 Queue-length cutoff model Depending upon the number of available servers server cutoff discipline determines which classes of patients are qualified for service The example shown in Figure

3 11 has two cutoffs N high and N low Obviously N high is equal to the total number of servers Low priority customers enter service only if fewer than N low ( N high ) servers are busy The purpose of this method is to reserve servers for high priorities Taylor and Templeton [2] studied two variants of a simple two-priority server cutoff model: one assumes high priority customers backlogged in queue while the other assumes they are lost if all servers are busy Schaack and Larson [3] extended the two-priority case to the T -priority problem (T 3) In a subsequent paper Schaack and Larson [4] derived waiting time distribution of each class for an extension of this model which assumes that customers require a random number of servers for service The queue-length cutoff priority queueing model can be regarded as the dual problem of the server cutoff model Instead of considering the number of available servers it manipulates the system based on the queue lengths In the example shown in Figure 12 a cutoff number L is set on the low priority queue The servers only process the high priority queue if the low priority queue length is less than or equal to L Once the threshold L is exceeded part of or all of the servers go to serve the low priority queue Gross and Harris [5] published solutions of expected queue length and expected waiting time for a special two-priority model which assumes the headof-the-line discipline (ie L = ) Miller [6] obtained the steady-state probabilities by a matrix-geometric method for the same model Recently Knessl Choi and Tier [7] derived the joint queue-length distribution as an integral for their dynamic two priority queue-length cutoff model Our work builds upon their research by developing methodologies to obtain the desired queue-length cutoff L in the preemptive resume case for the situation when the weights associated with customers in both queues are constant and the situation when these weights change linearly with the queue lengths In a disaster setting the weight signifies the importance associated with timely medical treatment of the patient 2 Model Formulation Customers are designated into two priority classes which are numbered as class-1 and class-2 so that the smaller the number the higher the priority The arrivals follow independent Poisson processes with rates λ 1 and respectively A single server processes both types of the customers with a mean rate µ In order to make the system stable we assume the stability condition as ρ 1 + ρ 2 < 1 where ρ 1 = λ 1 µ and ρ 2 = µ Let X(t) and Y (t) be the number of class-1 and class-2 customers in the system at time t respectively We consider the bivariate process {(X(t) Y (t)) t 0} with state space S = {(i j) : i j = } The steady-state probabilities are defined as p ij = P r{ in steady-state i class-1 customers and j class-2 customers in the system} The service discipline is controlled by the queue-length-cutoff policy A cutoff number L is set on the lower priority class If the number of customers in the lower priority queue is less than or equal to L only class-1 customers are served Once the threshold L is exceeded the server preempts the customer of class-1 currently in service The server keeps serving class-2 customers until the queue length of class- 2 is shortened to L Then the server preempts the class-2 customer who is being processed and switches back to service class-1 customers The sequence within each class is ordered on a first come first served basis When there is an empty queue the server only processes the other queue regardless of the threshold L This problem in summary is a Poisson-arrival exponential-service single-server two-priority queue with the preemptive queue-length-cutoff discipline 3

4 L-1 0 L 0 L+1 0 L+2 µ µ µ µ µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 λ L-1 1 L 1 L+1 1 L+2 µ µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 λ L-1 2 L 2 L+1 2 L+2 µ µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 µ λ 1 λ L-1 3 L 3 L+1 3 L+2 µ µ Fig 21 Rate Transition Diagram 3 Balance Equations and Generating Functions Our model is the same as that in [7] The rate transition diagram is shown in Figure 21 The system is separated into two main parts by the threshold L The first one gives class-1 customers higher priority while class-2 customers receive higher priority in the second one Equating flow in to flow out we get the balance equations for all sets of states in the dashed boxes in Figure 21 as follows: i = 0 and j = 0: (31) (λ 1 + )p 0 = µp 01 + µp 10 i = 0 and 1 j L 1: (32) (λ µ)p 0j = p 0j 1 + µp 0j+1 + µp 1j i = 0 and j = L: (33) (λ µ)p 0L = p 0 + µp 0L+1 + µp 1L i = 0 and j L + 1: (34) (λ µ)p 0j = p 0j 1 + µp 0j+1 i 1 and j = 0: (35) (λ µ)p i0 = µp i+10 + λ 1 p i 10 i 1 and 1 j L 1: (36) (λ µ)p ij = λ 1 p i 1j + p ij 1 + µp i+1j i 1 and j = L: (37) (λ µ)p il = λ 1 p i 1L + p i + µp i+1l + µp il+1

5 5 i 1 and j L + 1: (38) (λ µ)p ij = λ 1 p i 1j + p ij 1 + µp ij+1 In view of the difficulty in obtaining the solutions from the recursive method Knessl Choi and Tier [7] first derived the generating functions from the balance equations then got the probabilities by inverting the generating function Of interest in most applications are the measured system performances such as the expected number of class-1 customers N 1 and the expected number of class-2 customers N 2 in the system However their joint queue length is given by an integral which makes it difficult to calculate or even estimate N 1 and N 2 We calculate N 1 and N 2 by computing the first moment of the generating function To facilitate this we introduce the two-dimensional generating functions as: (39) H j (w) = p ij w i 0 j L 1 i=0 (310) H(w z) = p ij w i z j = z j H j (w) i=0 j=0 j=0 (311) G(w z) = p ij w i z j i=0 j=l (312) F (w z) = H(w z) + G(w z) 4 Expressions for Generating Functions Since the threshold L divides the system into two parts we need to calculate H(w z) and L(w z) separately to obtain the generating function F (w z) for the whole system We first consider H(w z) From equations (31) (32) (35) and (36) it is found that (41) H(w z) = ( µ w µ ) p 0 + z L H (w) + ( µ w µ ) p z 0j z j µz p 0L λ 1 w + z (λ µ) + µ w Details of this derivation are shown in Appendix A Similarly equations (33) (34) (37) and (38) yield [ λ 1 w + z (λ µ) + µ ] G(w z) z (42) ( 1 = z [ λ L 2 H (w) + µ z 1 w ) H L (w) + µ w p 0L Details of this derivation are shown in Appendix B By the method presented by Knessl Choi and Tier [7] the left-hand side of equation (42) can be rewritten as ] (43) z [(z z (w))(z z + (w))]g(w z)

6 6 where (44) z (w) = µ + λ 1 + λ 1 w (µ + λ 1 + λ 1 w) 2 4 µ 2 (45) z + (w) = µ + λ 1 + λ 1 w + (µ + λ 1 + λ 1 w) 2 4 µ 2 By setting z = z (w) we can get H L (w) in terms of H (w) and p 0L as (46) H L (w) = z (w) [ρ 2 wh (w) p 0L ] w z (w) Substituting equation (46) into equation (42) gives ( ) ( p0l ρ 2 wh (w) (47) G(w z) = ρ 2 (w z (w)) It follows that (48) z L z z + (w) F (w z) ( µ ) ( µ w µ p 0 + z L H (w) + w µ ) p z 0j z j µz p 0L = + λ 1 w + z (λ µ) + µ [ ] [ w p0l ρ 2 wh (w) z L ] ρ 2 (w z (w)) z z + (w) In order to evaluate the expression for F (w z) given in equation (48) p 0 p 0j (j = 1 L 1) H (w) and p 0L have to be determined We first focus on finding the initial state probability p 0 Intuitively for our problem p 0 is solely determined by µ λ 1 and and is not affected by the ordering of service Thus the probability of idleness should be the same as the one in the M/M/1 model with two input streams We formally establish this result in Proposition 41 Proposition 41 The idle probability is given by p 0 = 1 ρ 1 ρ 2 Proof Setting z = 1 in equations (41) and (47) we have ( ) 1 ( ) 1 H (w) + µ w 1 p 0j + µ w 1 p 0 + µp 0L H(w 1) = λ 1 w (λ 1 + µ) + µ w and ( ) ( ) p0l ρ 2 wh (w) 1 G(w 1) = ρ 2 (w z (w)) 1 z + (w) Then we set w = 1 in the equations above and use L Hospital s rule to get H(1 1) = µ p λ 0j + p 0L 2 and G(1 1) = µ ( ) p λ 0j + p 0L µ + p 2 µ λ 1 λ 0 2 )

7 By employing the condition that F (1 1) = 1 we find that p 0 = 1 ρ 1 ρ 2 Now we are going to describe how to calculate p 0j (j = 1 L 1) Define p (i j = 1 2 ) to be the state probabilities in the head-of-the-line case Miller [6] ij presented a series of recursive formulas for calculating p Knessl Choi and Tier [7] ij explained that p ij in our model are the same as the corresponding p for all i and ij 0 j L 1 Therefore Miller s method [6] can be directly used for our problem to obtain p 0j (j = 1 L 1) Our next focus is on deriving the expression for H (w) Equations (31) and (35) yield (49) [ λ 1 w (λ µ) + µ ] H 0 (w) = µp w 01 + µ ( 1 w w ) p 0 From equations (32) and (36) the relationship between H j (w) and H j 1 (w) is found as (410) [ λ 1 w (λ µ) + µ ] H j (w) + H j 1 (w) w = µp 0j+1 + µ w p 1 j L 1 0j 7 By setting A(w) = λ 1 w (λ µ) + µ and solving equations (49) and (410) w recursively we can establish the following result (presented without proof) Proposition 42 The general form of H j (w) (1 j L 1) is (411) H j (w) [ ( ) ( = µ 1 j 1 ) A(w) p A(w) + w 0j+1 + w (A(w)) j+1 (A(w)) j k 1 p 0j k ( ) k ( ) ] k=0 +( ) j 1 w w (A(w)) j+1 p 0 1 j L 1 By setting j = L 1 and w = 1 in equation (411) we get p 0L as (412) p 0L = ρ 2 H (1) Knessl Choi and Tier [7] presented an exact formula of H (1) as an integral However as they noticed for L > 30 the calculation becomes intractable Thus we use an approximate method to calculate H (1) From equation (39) we know that M H (1) = p i Thus H (1) is approximated by p i where M is a i=0 sufficiently large number in particular we will later see in Section 7 that using M = 5L works well under numerical tests 5 Derivation of Expected Numbers in System Armed with an expression of the generating function F (w z) we proceed to calculate L 1 and L 2 We take the partial derivatives of F (w z) in terms of both w and z and evaluate at (11) to get i=0

8 8 the results as (51) N 1 = 2µλ 1 p 0j + p 0 2µ H (1) (µ λ 1 )H (1) 2(µ λ 1 ) 2 + 2H (1) + H (1) 2(z (1) 1)(1 z +(1)) (H (1) + H (1))(z (1)) 2(1 z (1))2 (1 z + (1)) (H (1) + H (1))(z + (1)) (1 z (1))(1 z +(1)) 2 and (52) N 2 = (L 1)H (1) + 1 (j 1)p ρ 0j 2 [ H (1) + H + (1) ] [ ] L(1 z+ (1)) 1 z (1) 1 (1 z + (1)) 2 To evaluate N 1 and N 2 we observe that we further need to know the values of z (1) z (1) z + (1) z +(1) H (1) and H (1) These are as follows: z (1) = λ 1 µ z (1) = 2µλ2 1 (µ ) 3 z + (1) = µ z + (1) = µλ 1 (µ ) ( ) ( ) H (1) = µ λ1 µ λ1 + µ p 0L + and 2 2 p 0j + p 0 H (1) ( = µ 2µλ2 + 2(µ λ 1 ) 2 λ 3 2 ) p 0L + ( ) 2µλ2 + 2(µ λ 1 )(Lµ Lλ 1 + ) ( ) 2(µ λ1 )(λ 1 µ) ( ) + (j 1)p 0j 2(Lµ Lλ1 + ) λ 3 2 j=2 6 Properties Having studied the generating functions and derived the formulas for N 1 and N 2 we are ready to discuss some important properties of this queueing system As mentioned previously in Section 1 our queueing model is a generalization of the head-of-the-line model The first two properties are straightforward to establish The reader is referred to [8] for detailed proofs Property 61 When L = the queue-length cutoff model is reduced to the head-of-the-line model λ p 0 p 0j

9 The next property has been discovered through intuitive observation The point here is to investigate the mean number of customers (including both class-1 and class- 2) in the system If we consider the two classes as a whole it is instructive to point out that changing the value of L only changes the order of the service and it never changes the mean number of customers in the system Clearly the mean number of customers in our problem is the same as the one in the head-of-the-line model or even the same as the one in the nonpriority M/M/1 model It needs to be noted that the service rates of the two classes have been assumed to be equal and the classes have the same weight hence the class-1 and class-2 jobs are indistinguishable Property 62 Independent of the queue-length cutoff L the mean number of customers N 1 + N 2 is a constant which is given by 9 (61) N 1 + N 2 = λ 1 + µ λ 1 Although the mean total number of customers in the system is a constant it is quite natural to see that N 1 and N 2 do change as L changes Consider the example of increasing the value of L It is intuitively clear that the server spends more time on the high priority queue than before Thus N 1 decreases as L increases Conversely N 2 is an increasing function in terms of L Basically there are two different preemptive priority disciplines preemptive resume and preemptive repeat Preemptive resume allows a preempted customer to continue his/her service where he/she left off when he/she reenters service while preemptive repeat requires a preemptive customer to pick up a new value of service time from the service-time distribution whenever he/she reenters service The following property is only presented and proved under the first case ie preemptive resume Property 63 Under the preemptive resume priority discipline suppose there are two queue-length cutoffs L and L where L < L Then the following statements are true (62) N 1 (L) N 1 (L ) and (63) N 2 (L) N 2 (L ) Proof Since we are not even clear how H (1) and p 0j behave as L changes it seems impossible to prove this property directly by equations (51) and (52) The remarkable difficulty makes us resort to the following method We consider an arbitrary busy period Obviously when the priority discipline is preemptive resume the total service time of a class-1 or class-2 customer is in no way affected by the number of times he/she is preempted That is changing the value of L only changes the order of service while the duration of any busy period is always equivalent to the total service time of the customers in that period A typical example is shown in Figure 61 where the queue-length cutoff L = 4 It is instructive to see that N 1 and N 2 in a busy period can be calculated as (64) N i = Area(N L i ) i = 1 2 D

10 10 7 N t N 2 t Fig 61 An example of a busy period where Area(N L i ) = Area covered by class-i customers given that the queue-length cutoff is L and D = Duration of that busy period In this proof we only focus on the equation (62) The assertion of equation (63) can be derived in a similar manner Suppose the current queue-length cutoff is L If there is no preemption for class-1 customers in this period it is easy to see that the number of preemptions for class-1 customers is still zero if L is increased to L Consider now that there is at least one preemption for the high-priority queue We pick up an arbitrary preemption to study An example is shown in Figure 62 We can see that the preempted time point and the resume time point have been shifted from P L and R L to P L and R L respectively Clearly this shifting does not affect the area before time point P L Changes only occur after that time point Since the interarrival time between two customers and the service time of a customer cannot be zero the preemption time R L P L is strictly less than the one R L P L This result leads to another conclusion that at any time t ( P L ) X(t) in the second case is less than or equal to the corresponding quantity in the first case Thus when L is increased to L the area after time point P L is strictly decreased Combining the two areas together we can conclude that Area(N 1 ) is a decreasing function of L This yields the final result that N 1 (L) N 1 (L ) when L < L The property follows Property 62 tell us that the total number of customers in the system is constant However let us consider an example Suppose that there are 8 priority-1 customers and 2 priority-2 customers in case 1 and that there are 2 priority-1 customers and 8 priority-2 customers in case 2 Although the total number of customers is 10 for both cases it is obvious that case 1 is much worse than case 2 The reason is that

11 N 1 P L P L R L R L t Fig 62 An example of a preemption we usually assign a higher weight π 1 to priority-1 customers and a lower weight π 2 to priority-2 customers In consideration of the weighted number of customers in the system properties 62 and 63 lead us to the following result for minimizing the function π 1 N 1 + π 2 N 2 for certain choices of weights Property 64 In the preemptive resume model suppose that π 1 and π 2 are constant where π 1 > π 2 > 0 and π 1 + π 2 = 1 The optimal queue-length cutoff is given by L = Proof We consider two queue-length cutoffs L and L with L < L Suppose that the average number of priority-1 customers is N 1 and that the average number of priority-2 customers is N 2 if the cutoff is L Thus the weighted number of customers for this case is calculated as (65) π 1 N 1 + π 2 N 2 When L is increased to L property 63 tells us that N 1 decreases Suppose the number of priority-1 customers changes to N 1 ɛ where ɛ > 0 Then property 62 shows that the number of priority-2 customers changes to N 2 + ɛ The weighted number of customers is given by (66) π 1 N 1 + π 2 N 2 + (π 2 π 1 )ɛ It is easy to see that the value of equation (66) is smaller than the value of equation (65) Thus we conclude that the optimal cutoff is given by L = We now address the more interesting case which is particularly relevant to the disaster-relief application where the weight of a priority class may vary as the queue length changes Generally speaking the weight increases (decreases) as the queue length increases (decreases) Since π 1 and π 2 correlate each other (π 1 = 1 π 2 ) we only need to specify one of them eg π 2 We consider the case when the weight is a linear function of the queue length ie the function can be expressed as π 2 = KN 2 + C The function is shown in Figure 63 where N 2Lmin and N 2Lmax stand for the numbers of priority-2 customers under the minimal cutoff (L = 3) and the maximal cutoff (L = ) cases respectively The weights π 2Lmin and π 2Lmax for these two extreme cases are assumed to be given The parameters K and C are then determined uniquely by the two points (N 2Lmin π 2Lmin ) and (N 2Lmax π 2Lmax )

12 12 as follows: (67) K = π 2L max π 2Lmin N 2Lmax N 2Lmin and (68) C = π 2Lmax (π 2L max π 2Lmin )N 2Lmax N 2Lmax N 2Lmin π 2 1 π 2Lmax π 2L π 2Lmin 0 2L N N N N 2Lmin 2Lmax 2 Fig 63 A linear weight function The weighted number of customers for the minimal cutoff case is given by (69) π 1Lmin N 1Lmin + π 2Lmin N 2Lmin Assume that L is an arbitrary cutoff that is larger than L min Define δ and to be the values increased from π 2Lmin to π 2L and from N 2Lmin to N 2L respectively We can verify that δ = K Then the weighted number of customers for this case is calculated as (610) (π 1Lmin δ)(n 1Lmin ) + (π 2Lmin + δ)(n 2Lmin + ) which is equivalent to (611) (π 1Lmin N 1Lmin + π 2Lmin N 2Lmin ) +{2K 2 + [(π 2Lmin π 1Lmin ) + (N 2Lmin N 1Lmin )] } Comparing equation (611) with equation (69) we see that the optimal cutoff is determined by the discrete function (612) f( ) = 2K 2 + [(π 2Lmin π 1Lmin ) + (N 2Lmin N 1Lmin )]

13 13 Since K > 0 the value which minimizes the continuous equation (612) is (613) = (1 2π 2L min )(N 2Lmax N 2Lmin ) + (π 2Lmax π 2Lmin )(N 1Lmin N 2Lmin ) 4(π 2Lmax π 2Lmin ) However considering that 0 N 2Lmax N 2Lmin we can identify the following three cases: 1 Case 1: 0 In this case L = L min 2 Case 2: N 2Lmax N 2Lmin In this case L = L max 3 Case 3: 0 < < N 2Lmax N 2Lmin The function f( ) in our research is discrete Usually the optimal does not correspond to points in this discrete set In this case we only need to identify two points as follows: and 1 = min{ : f( ) f( )} 2 = min{ : f( ) > f( )} The optimal value of denoted as is given by = argmin{f( 1 ) f( 2 )} The cutoff L which corresponds to is the optimal solution 7 Computational Results Before proceeding with the numerical results we first investigate the approximate method of calculating H (1) As discussed in section 4 for L > 30 an appropriate value of M needs to used in order to M make i=0 p i as a good estimation of H (1) We conduct a series of numerical experiments using various combinations of λ 1 and We employ eight different values of L (from 3 to 10) in each experiment The exact results calculated by the method in [7] are used M as benchmarks After some trial runs we find that can provide a good i=0 p i approximation of H (1) if M = 5L In most of the cases the errors are within 01% Table 71 shows a sample of results from these experiments Next we focus on calculating N 1 and N 2 The results are shown in Figure 71 We can see that for all cases our model approaches the head-of-the-line case as L increases The total number of customers in the system is a constant while N 1 and N 2 decrease and increase respectively We now present an example which calculates the weighted number of customers under the case of a linear weight function We set π 2Lmin = 045 and π 2Lmax = 065 The results are shown in Table 72 We consider seven different combinations of λ 1 and In the case of λ 1 = = 01 since (= ) is less than zero we have L = L min The values of in all the other cases satisfy the condition 0 < < N 2Lmax N 2Lmin thus different finite optimal cutoffs are obtained as shown in Table 72

14 14 L Exact method Approximate method Error E E E E E E E E-05 Table 71 An example of the experiments with λ 1 = 03 = 02 and µ = 1 Fig 71 Computational results of N 1 N 2 and N 1 + N 2 Figure 72 also illustrates the detailed results When both λ 1 and are small (λ 1 = = 01) the change of the average weighted number of customers is negligible with a change in the value of L This is because the system is in an unsaturated status which leads to very little change of N 1 and N 2 values When either λ 1 or (or both) increases the average number of customers becomes much larger than in the unsaturated case We focus on the three cases (λ 1 = 01 = 08; λ 1 = 045 = 045; λ 1 = 08 = 01) in which the total arrival rates are the same We can see that the optimal cutoff increases as λ 1 increases Given the condition that λ 1 + is a constant property 62 shows that N 1 + N 2 is a constant no matter what the specific values of λ 1 and are Thus the increase of λ 1 causes an increase in N 1 and a decrease in N 2 More priority-1 customers lead to an increase in the

15 15 Weighted no of customers λ 1 = 01 = λ 1 = 01 = λ 1 = 045 = λ 1 = 08 = λ 1 = 01 = λ 1 = 049 = λ 1 = 088 = Table 72 Optimal cutoffs L optimal cutoff value As the system approaches the saturated status both N 1 and N 2 increase dramatically Consequently the average number of customers also increases significantly This causes a sharp increase in the optimal cutoff value Fig 72 Computational results of weighted number of customers 8 Summary and Future Work A two-priority preemptive single-server system with a queue-length cutoff queueing discipline has been studied in this paper This is a generic problem for various applications such as disaster relief and telecommunication Expressions for calculating the number of class-1 customers and the number of class-2 customers are developed based on a generating function approach The method we present does not lead to a closed-form solution but rather to an effective numerical approximation We have shown that our model reduces to the head-of-the-line model if L =

16 16 The total number of customers in the system is shown to be constant with respect to L Then we focus on the preemptive resume case in which N 1 and N 2 are decreasing and increasing functions of L respectively The weighted average number of customers is first analyzed for the case where the weights for both queues are constant We prove that the optimal policy is to set L = Then the case where the weights change linearly with the queue lengths is analyzed and a procedure is developed to find the optimal cutoff Numerical results illustrate the properties and other results There are several possible directions for future work: (1) In our model the moment that the number of low priority jobs hits threshold L the server stops working on high priority jobs entirely An alternative threshold policy in which the server is shared when the threshold is reached should be studied (2) For analytical tractability we assumed that the service rate for the high and low priority jobs is the same The more realistic case where the service rates are class dependent should be studied (3) Another direction of future work is to consider the use of an alternate solution method namely dimensionality reduction for Markov chains The work of Osogami Harchol-Balter and Scheller-Wolf [9] serves as a useful starting point (4) A further opportunity is in analyzing the multi-server version of our model which is closer to reality for a disaster-relief application (5) By applying the memoryless property of the Exponential distribution it may be possible to establish Property 63 for the preemptive repeat case Acknowledgements: This paper is supported by a grant from the Air Force Office of Scientific Research Grant No F The authors are grateful to constructive comments from two anonymous referees that led to much tighter presentation of the results

17 17 Appendix A: Derivation of equation (41) From equation (310) we get (1) (λ µ)h(w z) = (λ µ) p 0 + p 0j z j + p i0 w i + p ij w i z j i=1 i=1 From equations (31) (32) (35) and (36) the right-hand side of equation (1) can be written as (2) (λ µ)p 0 + ( p 0j 1 + µp 0j+1 + µp 1j )z j + + (λ 1 p i 1j + p ij 1 + µp i+1j )w i z j i=1 (µp i+10 + λ 1 p i 10 )w i i=1 Regrouping the terms in equation (2) according to λ 1 and µ we have (3) (λ µ)h(w z) = λ 1 p 0 + p i 10 w i + p i 1j w i z j i=1 i=1 + p 0 + p 0j 1 z j + p ij 1 w i z j i=1 +µ p 0 + (p 0j+1 + p 1j )z j + p i+10 w i + p i+1j w i z j i=1 i=1 Then we arrange the terms on the right-hand side in equation (3) and obtain (4) (λ µ)h(w z) = λ 1 wh(w z) + (zh(w z) z L H (w)) + µ ( µ w H(w z) + z µ ) p w 0j z j + µp 0 µ w p + 0 µz p 0L Equation (4) immediately yields (5) H(w z) = ( µ w µ ) p 0 + z L H (w) + ( µ w µ ) p z 0j z j µz p 0L λ 1 w + z (λ µ) + µ w Appendix B: Derivation of equation (42) From equation (311) we get (6) (λ µ)g(w z) = (λ µ) p 0L z L + p il w i z L + p 0j z j + p ij w i z j i=1 j=l+1 i=1 j=l+1

18 18 From equations (33) (34) (37) and (38) the right-hand side of equation (6) can be written as (7) z L ( p 0 + µp 0L+1 + µp 1L ) + z L (λ 1 p i 1L + p i + µp i+1l + µp il+1 )w i + j=l+1 ( p 0j 1 + µp 0j+1 )z j + i=1 i=1 j=l+1 (λ 1 p i 1j + p ij 1 + µp ij+1 )w i z j Regrouping the terms in equation (7) according to λ 1 and µ we have (8) (λ 1 + λ 2 + µ)g(w z) i 1 + = λ 1 w z L p i 1L w p i 1j w i 1 z j i=1 + z z p 0 + i=1 j=l+1 p 0j 1 z j 1 + z j=l+1 i=1 p i w i + p ij 1 w i z j 1 + µ z L+1 (p z 0L+1 + p 1L ) + i=1 j=l+1 +z L+1 (p i+1l + p il+1 )w i + p ij+1 w i z j+1 i=1 i=1 j=l+1 j=l+1 p 0j+1 z j+1 Then we arrange the terms on the right-hand side in equation (8) and obtain (9) (λ µ)g(w z) = λ 1 wg(w z) + zg(w z) + z L H (w) + µ ( [µ z G(w z) + zl 1 z H L(w) + 1 ) w H L(w) µ ] w p 0L Equation (9) immediately yields [ (10) λ 1 w + z (λ µ) + µ ] G(w z) ( z 1 = z [ λ L 2 H (w) + µ z 1 w ) H L (w) + µ w p 0L ] REFERENCES [1] N M Al-Momani and J R Harrald Sensitivity of earthquake loss estimation model: how useful are the predictions International Journal of Risk Assessment and Management Vol 4 No 1 (2003) pp 1 19 [2] ID S Taylor and J G C Templeton Waiting time in a multi-server cutoff-priority queue and its application to an urban ambulance service Opns Res Vol 28 No 5 (1980) pp [3] C Schaack and R C Larson An N-server cutoff priority queue Opns Res Vol 34 No 2 (1986) pp [4] C Schaack and R C Larson An N server cutoff priority queue where arriving customers request a random number of servers Mangt Sci Vol 35 No 5 (1989) pp [5] D Gross and C M Harris Fundamentals of queueing theory Wiley New York 1998 [6] D R Miller Computation of steady-state probabilities for M/M/1 priority queues Opns Res Vol 29 No 5 (1981) pp

19 [7] C Knessl D I Choi and C Tier A dynamic priority queue model for simultaneous service of two traffic types SIAM J Appl Math Vol 63 No 2 (2002) pp [8] Q Gong Responding to casualties in a disaster relief operation: initial ambulance allocation and reallocation and switching of casualty priorities PhD dissertation Department of Industrial and Systems Engineering University at Buffalo (SUNY) Buffalo NY 2005 [9] T Osogami M Harchol-Balter and A Scheller-Wolf Analysis of cycle stealing with switching cost Proceedings of the 2003 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems San Diego CA (2003) pp