Flexibility in an asymmetric system with prolonged service time at non-dedicated servers
aa r X i v : . [ c s . PF ] J u l The prolonged service time at non-dedicated servers in a poolingsystem
Yanting Chen a , Jingui Xie b , Taozeng Zhu c, ∗ a School of Mathematics, Hunan University, Changsha, China 410082 b School of Management, University of Science and Technology of China, Hefei, China 230026 c School of Computing, National University of Singapore, Singapore 117417
Abstract
In this paper, we investigate the e ff ect of the prolonged service time at the non-dedicated serversin a pooling system on the system performance. We consider the two-server loss model withexponential interarrival and service times. We show that if the ratio of the mean service time atthe dedicated server and the mean prolonged service time at the non-dedicated server exceeds acertain threshold, pooling would become unfavourable. In particular, the threshold is explicitlyprovided. Moreover, when the degree of the prolonged service time is pre-specified, we showthat the pooling system with prolonged service time at non-dedicated servers is not preferredwhen the work load in the system is greater than a certain threshold. Keywords:
Prolonged service time, Pooling
1. Introduction
Empirical studies indicate that the hidden (negative) consequences exist when the customeris served at the non-dedicated server. For instance, when the patients are assigned from a wardwhose designated beds are fully occupied to an available bed in a unit designated for a di ff erentservice, it has been shown in [1] that this ‘o ff -service placement’ is associated with a substantialincrease in remaining hospital length of stay, i.e., a prolonged service time. Therefore, it isof interest to investigate the e ff ect of such prolonged service time at non-dedicated servers ina pooling system on the system performance. In particular, it is desirable to have quantitativeresults that capture the relationship between the degree of the prolonged service time at non-dedicated servers and the system performance. This research is not only of practical importancebut also provides a theoretical attempt to understand the e ff ect of the prolonged service time atnon-dedicated servers in a pooling system on the system performance. We adopt a queueingapproach. To the best of our knowledge, our attempt is the first to model and analyze this newlyemerged e ff ect in pooling thoroughly. ∗ Corresponding author
Email addresses: [email protected] (Yanting Chen), [email protected] (Jingui Xie ), [email protected] (Taozeng Zhu )
Preprint submitted to Elsevier July 6, 2020 . Problem description and model formulation
We focus on the two-server loss system, i.e., there is no waiting space in front of the serversand the arriving customer who finds a busy server would be lost immediately. The dedicatedservers for Type 1 and Type 2 customers are Server 1 and Server 2 respectively. We first describethe independent system, and we assume that the arrival processes of Type 1 and Type 2 customersare two independent Poisson processes with the same parameter λ . When a Type 1 (Type 2)customer finds Server 1 (Server 2) is busy upon arriving, this customer will leave immediately.No pooling is allowed here. The service time for a Type 1 customer at Sever 1 is exponential withparameter µ and the service time for a Type 2 customer at Server 2 is exponential with parameter µ as well. We assume that λ > µ >
0. The ratio defined as the occupation rate ρ = λ/µ satisfies ρ >
0. The loss system is always stable, see [2]. Therefore, the occupation rate ρ is notrequired to be less than 1 here. The arrivals and services are mutually independent. The modelis depicted in Figure 1. µµλλ Type 1 customersType 2 customers Server 1Server 2
Figure 1: The independent system.
Before describing the pooling system, we first explain the prolonged service time at non-dedicated servers by introducing the prolonged coe ffi cient which is denoted by γ . The servicerate of any customer at the non-dedicated server is γ times the service rate of this customer at thededicated server. Due to the ine ffi ciency of dealing with non-dedicated customers, the prolongedcoe ffi cient γ satisfies 0 ≤ γ ≤ γµ .This characterization allows us to define the following pooling system with prolonged servicetime at non-dedicated servers.We now describe the pooling system with prolonged service time at non-dedicated servers.The arrival processes of Type 1 and Type 2 customers are two independent Poisson processeswith the same parameter λ . When a Type 1 customer finds Server 1 is busy upon arriving, thisType 1 customer would immediately go to Server 2, if Server 2 is idle at this moment, this Type1 customer would receive service at Server 2, otherwise, this Type 1 customer would leave thesystem immediately. The behaviour of the Type 2 customer is defined similarly. The service timefor a Type 1 customer at Sever 1 is exponential with parameter µ and at Server 2 is exponentialwith parameter γµ . Similarly, the service time for a Type 2 customer at Server 2 is exponentialwith parameter µ and at Server 1 is exponential with parameter γµ . Recall the assumption that λ > µ >
0. The arrivals and services are mutually independent. The model is depicted inFigure 2. 2 , γµµ , γµλλ Type 1 customersType 2 customers Server 1Server 2
Figure 2: The pooling system with prolonged service time at non-dedicated servers where 0 ≤ γ ≤ Due to the Poisson and the exponential assumptions, this pooling system with prolongedservice time at non-dedicated servers is a continuous-time Markov chain which can be denotedby X ( t ) = { ( i ( t ) , j ( t ) } where i ( t ) , j ( t ) ∈ { , , } for t ≥
0. When i ( t ) =
0, the Server 1 is emptyat time t . When i ( t ) =
1, the Server 1 is serving a Type 1 customer at t . When i ( t ) =
2, theServer 1 is serving a Type 2 customer at time t . When j ( t ) = , ,
2, the Server 2 is empty,serving a Type 1 customer and serving a Type 2 customer at time t , respectively. We know thatthe continuous-time Markov chain X ( t ) is irreducible and positive recurrent, then the stationaryprobabilities which are denoted by π ( i , j ) where i , j ∈ { , , } exist. In particular, the stationaryprobabilities of X ( t ) are displayed in the next theorem. Theorem 1.
The stationary probabilities of the continuous-time Markov chain X ( t ) are π (0 , = γ + γ + γ ργ ( ρ + + γ ( ρ + + ρ ( ρ + γ ) + γρ ( ρ + γ ) ,π (0 , = ρ γ + γ + γρ π (0 , , π (0 , = ρ + ( γ + ργ + ρ + π (0 , ,π (1 , = ρ + ( γ + ργ + ρ + π (0 , , π (1 , = ρ + γρ γ + γ + γρ π (0 , ,π (1 , = ρ + ( γ + ρ γ + ρ + π (0 , , π (2 , = ρ γ + γ + γρ π (0 , ,π (2 , = ρ γ + γ + γ ρ π (0 , , π (2 , = ρ + γρ γ + γ + γρ π (0 , , where ρ = λµ . roof . The stationary probabilities must satisfy the following balance equations2 λπ (0 , = µπ (1 , + γµπ (0 , + µπ (0 , + γµπ (2 , , (2 λ + γµ ) π (0 , = µπ (1 , + γµπ (2 , , (2 λ + µ ) π (0 , = λπ (0 , + µπ (1 , + γµπ (2 , , (2 λ + µ ) π (1 , = λπ (0 , + γµπ (1 , + µπ (1 , , ( µ + γµ ) π (1 , = λπ (1 , + λπ (0 , , µπ (1 , = λπ (0 , + λπ (1 , , (2 λ + γµ ) π (2 , = γµπ (2 , + µπ (2 , , γµπ (2 , = λπ (2 , + λπ (0 , , ( γµ + µ ) π (2 , = λπ (2 , + λπ (0 , . Recall that ρ = λµ , the balance equations can be rewritten in terms of γ and ρ ,2 ρπ (0 , = π (1 , + γπ (0 , + π (0 , + γπ (2 , , (2 ρ + γ ) π (0 , = π (1 , + γπ (2 , , (2 ρ + π (0 , = ρπ (0 , + π (1 , + γπ (2 , , (2 ρ + π (1 , = ρπ (0 , + γπ (1 , + π (1 , , (1 + γ ) π (1 , = ρπ (1 , + ρπ (0 , , π (1 , = ρπ (0 , + ρπ (1 , , (2 ρ + γ ) π (2 , = γπ (2 , + π (2 , , γπ (2 , = ρπ (2 , + ρπ (0 , , ( γ + π (2 , = ρπ (2 , + ρπ (0 , . Moreover, the normalization requirement is P i = P j = π ( i , j ) =
1, it can be readily verified thatthe stationary probabilities proposed in the theorem satisfy the balance equations and the nor-malization requirement, which completes the proof.We note that only the ratio of λ to µ , which is denoted by ρ and the prolonged coe ffi cient γ are relevant for the following analysis, where the rest of the parameters above are left out.
3. The blocking probabilities
The blocking probabilities are used to evaluate the performances of the independent systemand the pooling system with prolonged service time at non-dedicated servers. We first demon-strate the blocking probabilities here.In the independent system, the blocking probability (see [2]) for each type of customer, whichis denoted by P , is P = ρρ + ρ = λµ . In the pooling system with prolonged service time at non-dedicated servers, using PASTAproperty (Possion Arrivals See Time Averages, see [3, 4]), the blocking probability, which is4enoted by P , for either type of customers is π (1 , + π (1 , + π (2 , + π (2 , P is P = ρ ( γ + γ + γ ρ + γρ + ρ ) / ( γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ) . In the rest of the analysis, we first compare the performances of the independent system andthe pooling system with prolonged service time at non-dedicated servers for the fixed occupationrate ρ and di ff erent prolonged coe ffi cient γ , then we compare the performances of the indepen-dent system and the pooling system with prolonged service time at non-dedicated servers for thepre-fixed prolonged coe ffi cient γ where the occupation rate ρ is allowed to change.
4. Condition for pooling with prolonged service time at non-dedicated servers for fixed ρ In this section, we first investigate the monotonicity property of the blocking probability P for fixed ρ , then we provide the condition under which the pooling system with prolonged servicetime at non-dedicated servers is preferred when ρ is fixed. for fixed ρ For fixed ρ , we investigate the property of the blocking probability P while γ changes. In thenext lemma, we investigate the monotonicity of P considered as a function of γ and the extremevalues of P when γ = γ =
1, which are denoted by P γ = and P γ = respectively. Lemma 2.
For fixed ρ , the blocking probability P is monotonically decreasing in γ for γ ∈ [0 , . Moreover, we have P γ = ≥ P and P γ = ≤ P where P = ρρ + . P roof . Recall that P = ρ ( γ + γ + γ ρ + γρ + ρ ) / ( γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ) . For fixed ρ , let u ( γ ) and v ( γ ), which are functions of γ , denote the nominator and denominatorof P respectively. Specifically, we have u ( γ ) = ρ ( γ + γ + γ ρ + γρ + ρ ) and v ( γ ) = γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ . For fixed ρ , let L ( γ ) denote thenominator of the first derivative of P regarding to γ , we have L ( γ ) = u ′ ( γ ) v ( γ ) − u ( γ ) v ′ ( γ ) = − ρ (3 γ ρ + γ + γ ρ + γ ρ + γ ρ + γ ρ + γ ρ + γρ + γρ + γρ + ρ ) . Because the parameter ρ is positive, it can be readily verified that L ( γ ) is non-positive when γ ∈ [0 , P , which is v ( γ ), is positive, we concludethat for fixed ρ , the blocking probability P is monotonically decreasing in γ for γ ∈ [0 , γ =
0, we have P γ = = ρ /ρ =
1, this indicates that P γ = ≥ P where P = ρρ + . When γ =
1, the inequality P γ = ≤ P holds immediately using Theorem 1 in [5], which completes theproof. 5his monotonicity result holds intuitively because when the prolonged coe ffi cient γ becomessmaller, the service time at the non-dedicated server becomes longer, which would lead to moreblocking in the system. ρ For fixed ρ , we compare the blocking probability of the independent system, P , with theblocking probability of the pooling system with prolonged service time at non-dedicated servers, P . When P < P , the independent system is preferred, when P > P , the pooling systemwith prolonged service time at non-dedicated servers is preferred. Denote the di ff erence of theblocking probabilities P and P by g ( γ ), we have g ( γ ) = P − P = ρ/ ( ρ + − ρ ( γ + γ + γ ρ + γρ + ρ ) / ( γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ) = g n ( γ ) / g d ( γ ) . where g n ( γ ) = ρ ( γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ) − ρ ( γ + γ + γ ρ + γρ + ρ )( ρ +
1) and g d ( γ ) = ( ρ + γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ).To compare P and P , it is crucial to obtain the roots of g ( γ ) =
0, especially the potentialroot(s) in [0 , ρ , we now consider g ( γ ), g n ( γ ) and g d ( γ ) as functions of γ where γ is allowed to change from 0 to 1. Recall that ρ is positive and the prolonged coe ffi cient γ isrestricted to [0 , g d ( γ ) >
0. This indicates that it is su ffi cient to investigate theroot(s) in [0 ,
1] of g n ( γ ) = ,
1] of g ( γ ) =
0. Apparently,for fixed ρ , the nominator g n ( γ ) is a cubic function of γ . We present the roots of g n ( γ ) = Lemma 3.
For fixed positive ρ , the equation g n ( γ ) = has a unique root for γ ∈ [0 , . Inparticular, this root is γ = ρρ + . Moreover, the other two roots of g n ( γ ) = are γ = − and γ = − ρ . P roof . Recall the expression for g n ( γ ), we have g n ( γ ) = ρ ( γ ρ + γ ρ + γ + γ ρ + γ ρ + γ ρ + γ + γρ + γρ + ρ ) − ρ ( γ + γ + γ ρ + γρ + ρ )( ρ + = ρ ( ρ + γ − ρρ + γ + γ + ρ ) . Therefore, when ρ >
0, the roots of g n ( γ ) = γ = ρρ + , γ = − γ = − ρ , whichcompletes the proof.Based on the unique root of g n ( γ ) = γ ∈ [0 , γ and the ordering of the blockingprobabilities P and P . 6 heorem 4. For fixed ρ , we have P < P for γ ∈ [0 , ρρ + ) and P > P for γ ∈ ( ρρ + , . P roof . We know from Lemma 3 that the unique root of g n ( γ ) = ,
1] is ρρ + .Moreover, the expression g d ( γ ) is positive when ρ > γ ∈ [0 , γ = ρρ + is the unique root of g ( γ ) = P − P = , ρ the blocking probability P is monotonically decreasing in γ for γ ∈ [0 ,
1] and P γ = ≥ P , P γ = ≤ P . Moreover, the blocking probability P is a constant for fixed ρ . Weconclude that P < P for γ ∈ [0 , ρρ + ) and P > P for γ ∈ ( ρρ + ,
1] when ρ is fixed.Notice that Theorem 4 can also be proved using the expression for g n ( γ ) and the propertythat g d ( γ ) is positive for γ ∈ [0 ,
1] and ρ >
0. From Theorem 4, we see that the threshold ofthe prolonged coe ffi cient γ for allowing pooling with prolonged service time at non-dedicatedservers is precisely the blocking probability in the independent system, i.e., ρρ + . We also learnthat when the system gets busier (i.e., ρ ↑ ), the tolerance for the prolonged service time at thenon-dedicated server becomes lower as the interval for allowing pooling ( ρρ + ,
1] shrinks, seeFigure 3. Therefore, we conclude that the pooling with prolonged service time at non-dedicatedservers is suggested if the service rate at the non-dedicated server is greater than the service rateat the dedicated server times the blocking probability in the independent system. Otherwise, wesuggest the two servers to work separately. . . . . . . . . . . . . . . . . . . . . . Pooling system with prolonged service timeat non-dedicated servers Independent system ρ γ ρρ + Figure 3: The preferred service scheme for di ff erent ρ . In Figure 4, when ρ = the blocking probabilities P and P while γ changes are illustrated.We see that the blocking probability P increases when γ comes to 0. This observation holdsintuitively because when γ is small, the customer which has been assigned to the non-dedicatedserver would induce a very long service time, which would lead to more congestion in bothservers.
5. Condition for pooling with prolonged service time at non-dedicated servers for fixed γ Notice that if the prolonged coe ffi cient is given, i.e., γ is fixed, we are also able to provide thecondition under which the pooling system with prolonged service time at non-dedicated serversis preferred when the occupation rate, i.e., ρ , of the system changes.7 . . . . . . . . . . . . . . . . . γ B l o c k i ngp r ob a b iliti e s P P Figure 4: The blocking probabilities when ρ = . for fixed γ We first investigate the monotonicity of the blocking probability P for fixed γ in the nextlemma. Lemma 5.
For fixed γ ∈ [0 , , the blocking probability P is monotonically increasing in ρ for ρ > . P roof . Recall that P = ρ ( γ ρ + γρ + ρ + γ + γ ) / ( γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ ) . For fixed γ , let s ( ρ ) and t ( ρ ), which are functions of ρ , denote the nominator and denominatorof P respectively. Specifically, we have s ( ρ ) = ρ ( γ ρ + γρ + ρ + γ + γ ) and t ( ρ ) = γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ . For fixed γ , let Z ( ρ ) denote thenominator of the first derivative of P regarding to ρ , we have Z ( ρ ) = s ′ ( ρ ) t ( ρ ) − s ( ρ ) t ′ ( ρ ) = γρ (2 γ ρ + γ ρ + γρ + ρ + γ ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ ρ + γ ρ + γρ + γ + γ + γ )Because the prolonged coe ffi cient γ satisfies 0 ≤ γ ≤
1, it can be readily verified that Z ( ρ ) is non-negative for ρ >
0. Moreover, the square of the denominator of P , which is t ( ρ ), is positive, weconclude that for fixed γ , the blocking probability P is monotonically increasing in ρ for ρ > .2. The comparison of blocking probabilities for fixed γ For fixed γ , we again need to compare the blocking probabilities P and P when the occu-pation rate of the system, ρ is allowed to change. We now denote the di ff erence of the blockingprobabilities P and P by h ( ρ ), where h ( ρ ) = P − P = ρ/ ( ρ + − ρ ( γ ρ + γρ + ρ + γ + γ ) / ( γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ ) = h n ( ρ ) / h d ( ρ ) . where h n ( ρ ) = ρ ( γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ ) − ρ ( γ ρ + γρ + ρ + γ + γ )( ρ +
1) and h d ( ρ ) = ( ρ + γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ ).To compare P and P , it is crucial to obtain the roots of h ( ρ ) =
0, especially the potentialpositive root(s). For fixed γ , we now consider h ( ρ ), s ( ρ ), t ( ρ ) as functions of ρ where ρ is positive.Recall that γ satisfies γ ∈ [0 , h d ( ρ ) >
0. This indicates that it is su ffi cientto investigate the positive root(s) of h n ( ρ ) = h ( ρ ) =
0. For fixed γ , it can be readily verified that the nominator h n ( ρ ) is a cubic function of ρ (the coe ffi cient for ρ is 0), we present the roots of h n ( ρ ) = γ , γ =
1, i.e., there is no prolonged service time, has beenextensively studied in [5].
Lemma 6.
For fixed γ ∈ [0 , , the equation h n ( ρ ) = has a unique root for ρ ∈ (0 , ∞ ) . Inparticular, this root is ρ = γ − γ . Moreover, the other two roots of h n ( ρ ) = are ρ = and ρ = − γ . P roof . Recall the expression for h n ( ρ ), we have h n ( ρ ) = ρ ( γ ρ + γρ + ρ + γ ρ + γ ρ + γρ + γ ρ + γ ρ + γ + γ ) − ρ ( γ ρ + γρ + ρ + γ + γ )( ρ + = ( γ − ρ ( ρ − γ − γ )( ρ + γ ) . Therefore, for fixed γ ∈ [0 , h n ( ρ ) = ρ = γ − γ , ρ = ρ = − γ , whichcompletes the proof.Based on the unique positive root of h n ( ρ ) =
0, we are now ready to prove the theorem thatcharacterizes the relationship between the occupation rate ρ and the ordering of the blockingprobabilities P and P . Theorem 7.
For fix γ satisfying γ ∈ [0 , , we have P < P for ρ ∈ ( γ − γ , ∞ ) and P > P for ρ ∈ (0 , γ − γ ) . roof . We know from Lemma 6 that the unique positive root of h n ( ρ ) = γ − γ . Moreover, theexpression h d ( γ ) is positive when γ ∈ [0 ,
1) and ρ >
0. Therefore, we conclude that ρ = γ − γ isthe unique positive root of h ( ρ ) = P − P =
0. Recall that h n ( ρ ) = ( P − P ) h d ( ρ ) = ( γ − ρ ( ρ − γ − γ )( ρ + γ ) , we know that γ − < ρ > ρ + γ > γ ∈ [0 ,
1) and ρ >
0. Moreover, the expression h d ( ρ ) is positive for γ ∈ [0 ,
1) and ρ >
0. Therefore, we conclude that P < P for ρ ∈ ( γ − γ , ∞ )and P > P for ρ ∈ (0 , γ − γ ).From Theorem 7, we see that if the prolonged coe ffi cient is pre-fixed, the pooling systemwith prolonged service time is not encouraged when the system gets busier, see Figure 5. InFigure 6, for γ = , the blocking probabilities P and P when the occupation rate ρ increasesare illustrated. We see that when the work load in the system is light, the pooling system withprolonged service time at non-dedicated servers would lead to less blocking compared with theindependent system. However, when the system becomes rather busy, the blocking probability P will overtake the blocking probability P . Hence, it is suggested to keep the services independentwhen the work load in the system is heavy. . . . . . . . . . . . . Independent system Pooling system withprolonged service timeat non-dedicated servers γ ρ γ − γ Figure 5: The preferred service scheme for di ff erent γ .
6. Conclusion and discussion
Based on the two-server loss queueing model, we have investigated the e ff ect of the pro-longed service time at non-dedicated servers in a pooling system on the system performance.In particular, we have compared the blocking probabilities in the independent system and thepooling system with prolonged service time at non-dedicated servers.When the occupation rate ρ is fixed, we show that only when the prolonged coe ffi cient γ isgreater than a certain threshold, the pooling system with prolonged service time at non-dedicatedservers is preferred. More precisely, this threshold is the blocking probability in the independentsystem. When the prolonged coe ffi cient γ is fixed, we show that only when the occupation rate ρ is less than γ − γ , the pooling system with prolonged service time at non-dedicated servers ispreferred. Moreover, we have also demonstrated the monotonicity properties for the blocking10 . . . . . . . . . . . . . . . . . . . . ρ B l o c k i ngp r ob a b iliti e s P P Figure 6: The blocking probabilities when γ = . probability in the pooling system with prolonged service time at non-dedicated servers when theoccupation rate ρ is fixed or the prolonged coe ffi cient γ is fixed.The theoretical results suggest that the pooling with prolonged service time at non-dedicatedserver becomes unfavourable when the delay of the service at the non-dedicate server becomestoo substantial or the system is too busy. In future work, it will be of interest to generalize ourwork to incorporating the prolonged service time at non-dedicated servers in a queueing systemwith bu ff er space, multiple servers and asymmetric partners. Acknowledgement
Yanting Chen acknowledges support through the NSFC grant 71701066 and the FundamentalResearch Funds for the Central Universities.
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